Artificial intelligence first emerged as a term when John McCarthy coined it in his book What is Artificial Intelligence in 1956; defining it as the science and engineering of making intelligent machines . Ever since AI has been a domain of intense research and study. What researchers have been looking for in the capabilities of intelligent machines is knowledge, the ability to communicate, predict, reason, and other traits like planning, learning, and perception.
Artificial intelligence was more an issue of computer intelligence enhancement until it found a massive and dynamic field of robotics. Robots are machines that are designed to execute certain defined actions. They have arms and hands that follow certain instructions from a computer or a number of computers. The idea behind this is the researchers aim to design machines that possess the ability to move and manipulate objects. Strong AI, commonly known as General intelligence is a long-term goal of researchers. General intelligence includes everything that is humans. Their actions, perception, thinking, decision making, and execution; are some primary features.
The field of robotics now stands in close integration with AI. The bottom line is: robots ought to be human-like, and the foremost requirement for being human-like is near-to-human intelligence. Primarily, intelligence is required for robots to be able to handle such tasks as object manipulation and navigation, and secondly, localization; knowing location, mapping; learning what is around and motion planning; figuring out how to move and reach the target (Russel and Norvig 815-905).
The biggest challenge scientists are facing is the object affordances of the robots. An affordance is a quality of an object that allows a robot to perform an action. In the context of Human-Machine Interaction, affordance refers to those action possibilities that machines or robots perceive to interact with the human or other objects (Norman 34). The underline theory is similar to how humans perceive an object through sight. Humans obtain information in two ways: speech and visual communication. These two means are parallel and independent. The perception, communication, and decision-making of the human brain work through a process based on visual language.
A visual language is a set of practices by which images can be used to communicate concepts. When a human brain hears or sees something, a visual image is formed in the brain with its rough characters. The brain responds and acts correspondingly. For instance, to hold an apple, this is readily perceived in the human brain how many fingers should open up, and what is approximately the weight of an apple; i.e. the affordance of apple is predefined.
Similarly, machines act on basis of object affordances. The signals they receive in various forms are encoded into digital signals and led to the computing unit of the robot. There are various methods for motion planning and responsiveness of robots. Simple robots work on response signals based on sensors. The sensors emit signals; sound and infrared rays, and echoes of those signals determine the response actions.
In other words, they see their surroundings based on rays and sound echoing back. Such robots are not expensive to make and are widely used in universities. More advanced and expensive robots have now come to use visual imaging just like humans. The digital eyes of robots recognize an object with its size, geometry, and color in a three-dimensional representation of the world. Most of the robots use CCD (charge-coupled device) cameras as vision sensors (Arkin 250). This image is then sent to a computing unit where geometry is matched with its affordance. The robot then manipulates, handles, grips, or transforms the object according to affordance.
One such method is referred to as Visual Motion Planning. The method skips the step of transferring image features back to the robot pose and hence makes motion plans directly in the image plane (Zhang & Ostrowski 199-208). For object characterization for manipulation, object affordances are installed according to their shapes, geometry, color, size, and weight. For certain robots, visual imaging is the most important factor. In their experiments, Pavese and Bauxbaum observed that the robot, in presence of similar targets and distracters, selected the target object almost always on basis of color (Pavese & Bauxbaum 559).
Although the major way to encode the affordances is language-based information, lately imitation has also been a way to learning affordances. In this way, robot not only carries out actions as already perceived according to predefined affordances, but it also learns from the environment. The algorithm works on the perceived actions and motions in the surroundings. The robot recognizes objects and motions in its environment. Then it interacts with the object and learns about its motion about the principal axis. The robot is then able to repeat the observed action (Kopicki 14-15).
Although it enhances the movements and interaction with the environment, a major part of movements and object affordances are predefined. Another scenario is learning affordances through contact. The robot reaches the object in different directions and learns about its physical features through contact (Kopicki 26). Other methods commonly used for encoding affordances include listening by microphone and programming to define actions and perceptions. Machine language is used to define three or two-dimensional structures of target objects. A near-human general intelligence robot would embrace all methods of learning object affordances and would have all possible reactions defined. Such is the long-term goal of robotics.
Bibliography
Russell, Stuart J. & Norvig, Peter (2003), Artificial Intelligence: A Modern Approach (2nd ed.), Upper Saddle River, NJ: Prentice Hall.
Donald A. Norman, The Design of Everyday Things, London/New York: MIT Press, 2000.
Pavese, Antonella; Buxbaum, Laurel J. Action matters: The role of action plans and object affordances in selection for action. In , Visual Cognition, Volume 9, 2002 pp. 559-590(32) Psychology Press.
This essay deals with the importance of geometry in our daily life. An essay includes many points to highlight the importance of geometry. It also specifies why students need to study geometry and the benefits for students in life. This essay also includes real examples of geometry in our life.
Importance of geometry in our life
In everyday life, people are always surrounded by different spaces and different belongings, which are of different shapes. Our universe itself is consists of different planets and stars. All these have got different shapes and symbols. To be able to understand the wonder of the worlds shape and appreciate it, we need to be able to understand and have knowledge of spatial use. In other words, the areas related to space and the position, size and shape of things in it (10 shocking reasons why geometry is important in your life, n.d., para.2).
When one gets the idea regarding the relationship between different shapes and sizes, they can be better prepared to use those in daily lives. Here comes the importance of geometry. Geometry assists in having accurate measurements and relationships of different shapes. Geometry will increase ones spatial understanding. It is often that people think of basic shapes and sizes always, many people think well visually (Shape and space in geometry, 2010, para.8). To visualize something, it is very significant that it requires an understanding of geometry. Only with the help of geometry, one can think of any kind of shape in mind before making it real.
In the workplace also use of geometry is very important. Knowledge regarding geometry is very important in order to outshine in the work. The use of geometry gives exercise for the left and right sides of the brain. The left brain is more advanced in using technical and logical activities; at the same time, the right brain is very good at visualizing. Since geometry needs both, it provides very good brain exercise. In other words, geometry uses full use of the brain. Every man-made wonders that have been created in this world are with the help of geometry. It is with the help of geometry one is able to give life for his imaginative thinking. If geometry is not used, then everything will be in ones dream. All sorts of two and three-dimensional shapes that we see or come across are instigating in geometry. So, geometry is considered to be an unavoidable and very important part of human life.
Importance of geometry in students life
Geometry, the study of space and spatial relationships, is an important and essential branch of the mathematics curriculum at all grade levels. The ability to apply geometric concepts is a life skill used in many occupations (Geometry, n.d., para.1). Geometry is an excellent training ground for (Finkbeiner, 1995, p.54) all the students who need to make use of tangible experiments. Doing these types of the experiment will enrich their knowledge in the subjects. Many types of the mathematical experiment can be easily understandable by the use of geometry. Not only that with the help of geometry it is very easy for the students to gain their knowledge in different types of the experiment they are doing. By studying geometry, students can apply it to their real life. When students learn geometry, it always enhance logical reasoning (Jordan, n.d., para.3) and the thinking capability of the student. Developing logical reasoning and deductive thinking surely increases ones mental and mathematical ability. Development of these is very important in students as this will help in their career to achieve more and more. Not only in their career but in life also this studying of geometry will improve their thinking capacity. Understanding geometry will help students to take decisions properly and it will help them to find out solutions for problems they are facing in their life. It is certain that geometry students adequately develop their knowledge and skills for solving any kind of problem. (Dindyal, n.d., p.189).
Examples of geometry in real life
Thousands of examples can be shown for the use of geometry in our life. The use of geometry is inevitable in construction works. Before the beginning of the construction, architects draw the plan of the building using geometrical figures. The use of geometry in this field is not a new trend. It has been in use since the historic period itself. If you go back to Roman historical sites you will see such examples like the great coliseum. A great example can be seen is the famous Egyptian pyramid. Some other famous structures are Eiffel Tower which is in Italy, Chrysler in New York. If you look around your neighborhood house, you will see these shapes (How geometry is used in construction, 2010, para.2). Geometric principles are used by architects to ensure the safety of their constructions. In most of the legendary constructions of olden and new times, we can find smart use of basic geometrical principles. The new finding in these principles reflects the developments that have taken place in the building construction field.
Geometric rules are used in the medical field for the reconstruction of our inner and outer organs. Using the geometrical principle, human movement is analyzed for applying to the fields like Robotics. In constructing and controlling the movements of robots, it is very necessary to study human nature based on certain principles. We can relate different objects in the real world using geometry. In the computerized reconstruction of the real world, these principles are used. So, the principles of geometry play a big role. It is also used in graphic designing, video game creation, etc.
Geometry has a relevant role in astronomy also. In the systematic study of space and bodies in outer space, geometric principles are used. According to the Escher Math website, geometry allows astronomers to plan observations and reconstruct bodies in outer space such as asteroids. If gazing at the stars is something you enjoy, consider pairing your love of the night sky with your skills in geometry to become an astronomer (Hickman, 2010, para.6).
Many studies are going on to explore the mysteries of the universe. In all the studies in this direction, geometry has an important role. Studies have proven that it is possible that secret of the nature can be found by studying the links among geometric archetypes of different objects in nature. In nature, we find patterns, designs and structures from the most minuscule particles, to expressions of life discernible by human eyes, to the greater cosmos. These inevitably follow geometrical archetypes, which reveal to us the nature of each form and its vibrational resonances (Rawles, 2009, para.1).
Visual learning of Geometry
From the basic principles, geometry and its applications have developed a lot. Now, it has a vast area of application. To teach ideas of geometry, advanced study tools are necessary. It is almost impossible to learn 2D and 3D concepts of geometry without proper demonstrations. The highlights, interlaced with interactive demonstrations, are intuitively developed. By learning to recognize patterns and powerful knowledge discovery process evolved (Inselberg, n.d., para.1).
In order to learn different patterns, influential knowledge is required. It requires the help of geometrical concept. For example, recognition of M-dimensional objects form (M-1) requires lots to understand. For representing points in the plane, it is necessary that one should have knowledge regarding indices. This requires influential geometrical algorithm. That is, in order to make these algorithms application, knowledge is prerequisite. Applications of parallel coordinates include collision avoidance and conflict resolution algorithms for air traffic control (3 USA patents), computer vision (USA patent), data mining (USA patent) for data exploration and automatic classification, optimization, decision support and process control (Inselberg, n.d., para.3).
Conclusion
Geometry has got important role in life of the people, especially students. Geometry is considered to be important part of real life. Since world is built of shape and space, and geometry is its mathematics (Shape and space in geometry, 2010, para.5). Geometry is very helpful for the students in order to solve many problems. With the help of geometry, many students are presently solving many problems. This helps them to understand more. Finally, many people in the world are very well in thinking visually. In order to achieve this, geometry is considered to be doorway to achieve all the results. Students who are developing strong concept or intellect in the language of geometry can always excel in advanced topics related to mathematics. Thus, geometry is very important.
Reference
10 shocking reasons why geometry is important in your life. (n.d.). Math Worksheet Center. 2010. Web.
Dindyal, J. (n.d.). Algebraic thinking in geometry at high school level: Students use of variables an unknowns. Google docs. 2010. Web.
Finkbeiner, D.T. (1995). Recent publications. Mathematical Association of America, p.54. Web.
Geometry. (n.d.). Much More Math. 2010. Web.
Hickman, S. (2010). What types of jobs use geometry? eHow. Web.
How geometry is used in construction. (2010). Peerpapers.com. Web.
Inselberg, A. (n.d.). Parallel coordinates: Visual multidimensional geometry and its applications. 2010. Web.
Jordan, M. (n.d.). Why homework is important? Much More Math. 2010. Web.
Rawles, B.A. (2009). The geometry code: Symbolic wisdom of natural laws within us. Elysian Publishing. Web.
Shape and space in geometry. (2010). Annenberg Media. Web.
Mathematics is a very important subject because we use it in our day to day lives. Regardless of that, many learners express it as one of the most difficult subjects and that explains why many educators have been experiencing poor performance in this subject. This could be because most learners did not have a good foundation during their initial stages.
In mathematics, geometry is one of the most difficult subjects that pose many challenges to children. Children need to understand shapes, sizes, figures, and figures so as to appreciate geometry. This calls for proper foundation in geometrical concepts, both in schools and homes. Therefore, this paper will shed light on how educators can teach mathematics to children efficiently, particularly learning geometry.
According to Rich and Thomas (2008), the process of learning mathematics commences early enough even before the child reaches the age of going to school. But this study progresses automatically as the child gets acquainted to his or her surroundings. For instance, when you bring two toys to three children they will tell you that they are not enough and yet they do not know anything about numbers.
This is because they expect each one of them to have a toy. When a child is being introduced to mathematics, the teacher should start on a gradual pace by ensuring that the child first learns the basics. For instance you can never teach children how to add numbers when you have not taught them about numbers. This means that the basic lessons should come first.
Children gain knowledge through observation. Therefore, it would be important for the teacher to attract the attention of the child when he/she is demonstrating how the calculations are done. This can be achieved by asking questions at random to ensure that the childrens mind is glued to what is going on in the classroom. Moreover, asking questions helps the teacher to gauge the understanding of the learners (Clements, 2006).
If the teacher feels that a particular topic in mathematics was not well understood according to the performance of children in that topic, he/she should consider repeating that topic by using different approaches. Some of the methods that enhance understanding include selecting learners who understand the topic and have them demonstrate in front of the classroom how they were able to solve the sums.
The teacher should be present to make corrections where necessary. Above all, the teacher should be very patient when teaching children because their thinking capacity is still low and should consider asking questions about the things that were taught in the previous day before moving to another topic. This will help the teacher to identify the areas that need special attention.
Sarama and Clements (2006) explain that the teacher should pay special attention to all children without being limited to fast learners.
Besides, when the teacher does not engage children in his/her discussions, the childrens minds are most likely to be carried by other thoughts such as how they will watch the next cartoon episode. Moreover, listening in itself is a difficult task and that is why learners doze in class. This can be avoided by asking questions and also telling stories that relate to the topic being studied.
Mathematics is a very demanding subject hence the teacher should teach it when the kids are still fresh especially in the morning hours because in the afternoons the children are most likely to be exhausted. This is due to the fact that the time they spend on other subjects and as well as playing their games hence their level of concentration may decline.
Most teachers think that the best way to teach children mathematics is by giving them lengthy homework. This is very wrong because they may complete the assignments and yet they do not understand the concepts involved. In mathematics, the formula is the most vital element because unless the learner understands it their can be no answer to any mathematical problem.
It would be better if the child does a few sums that he/she understands than attempting a bulk of math that he/she does not understand. In such a case, the child will tackle the questions just to please the teacher and this may drive the child towards copying from peers which could continue to affect the child later in academic life.
The teacher should develop a habit of identifying slow learners in the classroom and keep an eye on their progress (Deiner, 2009).
Mathematics, especially geometry is best learnt through frequent exercises. This means that the child can be scheduled to solve four to five mathematical problems in a day. This goes a long way in preventing the situation where the childs mind is congested with lots of formulas that the child can hardly remember.
When children are being introduced to geometry, it is important to teach them first about the geometrical apparatus such as the divider and the protractor so that when they come across a geometrical set they know how to use every tool including the compass. In addition, the children should be taught about the various geometrical shapes such as the triangles and rectangles among many other shapes.
Brumbaugh, Ortiz and Grasham (2006) state that while teaching a tough topic like geometry the teacher should integrate the parents and guardians to ensure that even after the child is out of school the parents and guardians will continue teaching the same topic to the child indirectly.
The parent can make the child understand the topic better by making them apply the geometrical skills in their plays and with the things that they interact with the most. For instance, the parent can ask the child to measure the width and length of the television set.
Parents can integrate geometry into the games children play. This includes making the child ride the bicycle in circles. The child can also be asked to measure the distance covered while riding the bicycle within the home compound.
Besides, the parent can ask the child to identify different shapes in the television programs the child watches. In addition, the parent can make snacks in different shapes to help the child understand the shapes better.
Deiner (2009) outlines that in geometry, the childs understanding can be enhanced by displaying the various shapes and sizes in different pleasant colors. Besides, the teacher can also ask questions to the kid and provide assistance if the child gets stuck by giving a few hints towards the answer.
When the child works out a problem in the wrong way, the teacher should never give vague conclusions such as the answer was wrong or right but should rather elaborate the answer and help the child discover where he/she went wrong. This will make the child cautious about making the same mistake compared to when the teacher gives a vague remark.
Depending on the age of the child, the teacher can also employ arithmetic story books. This is in a bid to make the topic more interesting. The teacher needs to conduct assessment tests after covering a few areas of geometry.
The learners who achieve the highest marks should be rewarded with small gifts like cookies. Even without tests the teacher can motivate the children by requesting them to clap their hands for those that answer questions correctly.
Furthermore, children can be organized into small groups and then assigned problems to solve individually. In such case, the teacher should dig deeper into the childs understanding by seeking to find out how the child at his answer.
This is accomplished by asking the child to explain why he gave a particular answer. Both the teacher and the parent need to be friendly to the child because if they are hostile or give lecture like remarks when the child makes a mistake it may demoralize the child.
The teacher can put on a warm smile in the classroom while the parent can offer a bar of chocolate during home based learning sessions. Both educators should also use a polite tone while speaking to the child. This also includes correctly choosing the words to use. The child should be made to identify the objects in his surroundings that are in the shapes taught in geometry class.
This can be items like plates, cups and beds among others. The parent should constantly remind the child about geometry by asking questions frequently such as when the child holds an item that has a geometrical shape (Garfias, 2011).
During class discussions every child should be allowed to express his views because that way the children will learn something from each other.
Besides, sharing their thoughts will provide a room for correction and thus build the childs confidence while tackling such questions because he will remember what they learnt as a group. In some cases the children can be asked to write short essays about the topic. This practice aims at displaying their level of knowledge in the topic.
Harris and Turkington (2000) explain that practical exercises are also crucial in geometry because they enable children to demonstrate their skills. Such exercises can be carried out in a different location apart from the classroom such as in the play ground because they require more space for the shapes to be laid out.
The teacher can issue materials like blocks and porters mud and ask the kids to make the shapes they have learnt in class. Note that in this case there are no books to refer to.
In conclusion, geometry and mathematics in general should be made to look like a hobby for kids. If every child is provided with the appropriate guidance in understanding mathematics, the number of poor grades in science subjects that are reported in institutions of higher learning would diminish gradually because every learner would have changed his/her attitude.
Therefore, it is the duty of teachers and parents to assist children in learning mathematics.
References
Brumbaugh, K.D., Ortiz, E., & Gresham, G. (2006). Teaching Middle School Mathematics. New York: Routledge.
Clements, D. (2006). Ready for Geometry! From an Early Age, Children make Sense of the Shapes they see in the World around Them . International Journal of Mathematical Education, Science and Technology. 2: 5-6.
Deiner, L.P. (2009). Inclusive Childhood Education: Development, Resources and Practice. New Delhi: Cengage Learning.
Garfias, L.E. (2011). Literal Math for Little Minds . Whatever State I Am. Web.
Harris, J. & Turkington, C. (2000).Get ready! For Standardized Tests: Grade 2. New York: McGraw-Hill.
Rich, B. & Thomas, C. (2008). Schaums Outline of Geometry. New York: McGraw-Hill.
Sarama, J. & Clements, D.H. (2006). Early Math: Introducing Geometry to Young Children . Scholastic. Web.
Artificial intelligence first emerged as a term when John McCarthy coined it in his book “What is Artificial Intelligence” in 1956; defining it as “the science and engineering of making intelligent machines”. Ever since AI has been a domain of intense research and study. What researchers have been looking for in the capabilities of intelligent machines is knowledge, the ability to communicate, predict, reason, and other traits like planning, learning, and perception.
Artificial intelligence was more an issue of computer intelligence enhancement until it found a massive and dynamic field of robotics. Robots are machines that are designed to execute certain defined actions. They have arms and hands that follow certain instructions from a computer or a number of computers. The idea behind this is the researcher’s aim to design machines that possess the ability to move and manipulate objects. Strong AI, commonly known as General intelligence is a long-term goal of researchers. General intelligence includes everything that is humans. Their actions, perception, thinking, decision making, and execution; are some primary features.
The field of robotics now stands in close integration with AI. The bottom line is: robots ought to be human-like, and the foremost requirement for being human-like is near-to-human intelligence. Primarily, intelligence is required for robots to be able to handle such tasks as object manipulation and navigation, and secondly, localization; knowing location, mapping; learning what is around and motion planning; figuring out how to move and reach the target (Russel and Norvig 815-905).
The biggest challenge scientists are facing is the object affordances of the robots. An affordance is a quality of an object that allows a robot to perform an action. In the context of Human-Machine Interaction, affordance refers to those action possibilities that machines or robots perceive to interact with the human or other objects (Norman 34). The underline theory is similar to how humans perceive an object through sight. Humans obtain information in two ways: speech and visual communication. These two means are parallel and independent. The perception, communication, and decision-making of the human brain work through a process based on visual language.
A visual language is a set of practices by which images can be used to communicate concepts. When a human brain hears or sees something, a visual image is formed in the brain with its rough characters. The brain responds and acts correspondingly. For instance, to hold an apple, this is readily perceived in the human brain how many fingers should open up, and what is approximately the weight of an apple; i.e. the affordance of apple is predefined.
Similarly, machines act on basis of object affordances. The signals they receive in various forms are encoded into digital signals and led to the computing unit of the robot. There are various methods for motion planning and responsiveness of robots. Simple robots work on response signals based on sensors. The sensors emit signals; sound and infrared rays, and echoes of those signals determine the response actions.
In other words, they ‘see’ their surroundings based on rays and sound echoing back. Such robots are not expensive to make and are widely used in universities. More advanced and expensive robots have now come to use visual imaging just like humans. The digital eyes of robots recognize an object with its size, geometry, and color in a three-dimensional representation of the world. Most of the robots use CCD (charge-coupled device) cameras as vision sensors (Arkin 250). This image is then sent to a computing unit where geometry is matched with its affordance. The robot then manipulates, handles, grips, or transforms the object according to affordance.
One such method is referred to as Visual Motion Planning. The method skips the step of transferring image features back to the robot pose and hence makes motion plans directly in the image plane (Zhang & Ostrowski 199-208). For object characterization for manipulation, object affordances are installed according to their shapes, geometry, color, size, and weight. For certain robots, visual imaging is the most important factor. In their experiments, Pavese and Bauxbaum observed that the robot, in presence of similar targets and distracters, selected the target object almost always on basis of color (Pavese & Bauxbaum 559).
Although the major way to encode the affordances is language-based information, lately imitation has also been a way to learning affordances. In this way, robot not only carries out actions as already perceived according to predefined affordances, but it also learns from the environment. The algorithm works on the perceived actions and motions in the surroundings. The robot recognizes objects and motions in its environment. Then it interacts with the object and learns about its motion about the principal axis. The robot is then able to repeat the observed action (Kopicki 14-15).
Although it enhances the movements and interaction with the environment, a major part of movements and object affordances are predefined. Another scenario is learning affordances through contact. The robot reaches the object in different directions and learns about its physical features through contact (Kopicki 26). Other methods commonly used for encoding affordances include listening by microphone and programming to define actions and perceptions. Machine language is used to define three or two-dimensional structures of target objects. A near-human general intelligence robot would embrace all methods of learning object affordances and would have all possible reactions defined. Such is the long-term goal of robotics.
Bibliography
Russell, Stuart J. & Norvig, Peter (2003), Artificial Intelligence: A Modern Approach (2nd ed.), Upper Saddle River, NJ: Prentice Hall.
Donald A. Norman, The Design of Everyday Things, London/New York: MIT Press, 2000.
Pavese, Antonella; Buxbaum, Laurel J. Action matters: The role of action plans and object affordances in selection for action. In , Visual Cognition, Volume 9, 2002 pp. 559-590(32) Psychology Press.
The event that substantially contributed to the development of mathematics as a science was the flooding of the Nile. I specifically chose this event because the flooding takes place every year, emphasizing how the ancient communities were similar to the current society due to living in the same conditions. In ancient times the sciences were driven by practical concerns, such as measures of land and weight. Therefore, in the case of Nile floods, according to Herodotus, resurveying the fields required the application of measures and principles of geometry, which at those times was limited to earth measurements (Bertoloni et al., 2006).
To resurvey the land, Egyptians used ropes and measured the dimensions of land plots to accurately divide them for taxation purposes. The stretched rope with knots used to divide the rope into twelve equal parts was called a “harpedonaptai” (Rowe, 2018, p.243). The rope could form a 3-4-5 triangle which was used for building purposes to define foundations.
Regarding society’s reaction to the event and its social context, the flooding was acknowledged as one of the year’s most important events. As rainfalls are very rare in Egypt, the flooding was the only opportunity for the earth to maintain the moisture level required to grow the crops. Moreover, the rivers presented the center of development of civilizations, with Mesopotamia even being called a “land between rivers” (McClellan & Dorn, 2006, p.33). Ancient Egyptians’ calendar started from the flooding due to its influence and benefits that the flooding caused on Egypt’s agricultural and economic development. Thus, by resurveying the land after the flooding, the ancient Egyptians established the early principles of geometry. The event emphasizes the practical origins of geometry and mathematics and draws the connection to how geometry was used for economic and agricultural purposes.
References
Bertoloni, M. D., Dorn, H., & McClellan, J. E. I. (2006). Science and technology in world history: An introduction. Johns Hopkins University Press.
McClellan, J. E., & Dorn, H. (2006). Science and technology in world history: An introduction (2nd ed.). Johns Hopkins University Press.
Rowe, D. E. (2018) A Richer picture of mathematics. Springer.
This essay deals with the importance of geometry in our daily life. An essay includes many points to highlight the importance of geometry. It also specifies why students need to study geometry and the benefits for students in life. This essay also includes real examples of geometry in our life.
Importance of geometry in our life
In everyday life, people are always surrounded by different spaces and different belongings, which are of different shapes. Our universe itself is consists of different planets and stars. All these have got different shapes and symbols. “To be able to understand the wonder of the world’s shape and appreciate it, we need to be able to understand and have knowledge of spatial use. In other words, the areas related to space and the position, size and shape of things in it” (10 shocking reasons why geometry is important in your life, n.d., para.2).
When one gets the idea regarding the relationship between different shapes and sizes, they can be better prepared to use those in daily lives. Here comes the importance of geometry. Geometry assists in having accurate measurements and relationships of different shapes. Geometry will increase one’s spatial understanding. It is often that people think of basic shapes and sizes always, “many people think well visually” (Shape and space in geometry, 2010, para.8). To visualize something, it is very significant that it requires an understanding of geometry. Only with the help of geometry, one can think of any kind of shape in mind before making it real.
In the workplace also use of geometry is very important. Knowledge regarding geometry is very important in order to outshine in the work. The use of geometry gives exercise for the left and right sides of the brain. The left brain is more advanced in using technical and logical activities; at the same time, the right brain is very good at visualizing. Since geometry needs both, it provides very good brain exercise. In other words, geometry uses full use of the brain. Every man-made wonders that have been created in this world are with the help of geometry. It is with the help of geometry one is able to give life for his imaginative thinking. If geometry is not used, then everything will be in one’s dream. All sorts of two and three-dimensional shapes that we see or come across are instigating in geometry. So, geometry is considered to be an unavoidable and very important part of human life.
Importance of geometry in student’s life
“Geometry, the study of space and spatial relationships, is an important and essential branch of the mathematics curriculum at all grade levels. The ability to apply geometric concepts is a life skill used in many occupations” (Geometry, n.d., para.1). Geometry is “an excellent training ground for” (Finkbeiner, 1995, p.54) all the students who need to make use of tangible experiments. Doing these types of the experiment will enrich their knowledge in the subjects. Many types of the mathematical experiment can be easily understandable by the use of geometry. Not only that with the help of geometry it is very easy for the students to gain their knowledge in different types of the experiment they are doing. By studying geometry, students can apply it to their real life. When students learn geometry, it always “enhance logical reasoning” (Jordan, n.d., para.3) and the thinking capability of the student. Developing logical reasoning and deductive thinking surely increases one’s mental and mathematical ability. Development of these is very important in students as this will help in their career to achieve more and more. Not only in their career but in life also this studying of geometry will improve their thinking capacity. Understanding geometry will help students to take decisions properly and it will help them to find out solutions for problems they are facing in their life. It is “certain that geometry students adequately develop their knowledge and skills for solving” any kind of problem. (Dindyal, n.d., p.189).
Examples of geometry in real life
Thousands of examples can be shown for the use of geometry in our life. The use of geometry is inevitable in construction works. Before the beginning of the construction, architects draw the plan of the building using geometrical figures. The use of geometry in this field is not a new trend. It has been in use since the historic period itself. “If you go back to Roman historical sites you will see such examples like the great coliseum. A great example can be seen is the famous Egyptian pyramid. Some other famous structures are Eiffel Tower which is in Italy, Chrysler in New York. If you look around your neighborhood house, you will see these shapes” (How geometry is used in construction, 2010, para.2). Geometric principles are used by architects to ensure the safety of their constructions. In most of the legendary constructions of olden and new times, we can find smart use of basic geometrical principles. The new finding in these principles reflects the developments that have taken place in the building construction field.
Geometric rules are used in the medical field for the reconstruction of our inner and outer organs. Using the geometrical principle, human movement is analyzed for applying to the fields like Robotics. In constructing and controlling the movements of robots, it is very necessary to study human nature based on certain principles. We can relate different objects in the real world using geometry. In the computerized reconstruction of the real world, these principles are used. So, the principles of geometry play a big role. It is also used in graphic designing, video game creation, etc.
Geometry has a relevant role in astronomy also. In the systematic study of space and bodies in outer space, geometric principles are used. “According to the Escher Math website, geometry allows astronomers to plan observations and reconstruct bodies in outer space such as asteroids. If gazing at the stars is something you enjoy, consider pairing your love of the night sky with your skills in geometry to become an astronomer” (Hickman, 2010, para.6).
Many studies are going on to explore the mysteries of the universe. In all the studies in this direction, geometry has an important role. Studies have proven that it is possible that secret of the nature can be found by studying the links among geometric archetypes of different objects in nature. “In nature, we find patterns, designs and structures from the most minuscule particles, to expressions of life discernible by human eyes, to the greater cosmos. These inevitably follow geometrical archetypes, which reveal to us the nature of each form and its vibrational resonances” (Rawles, 2009, para.1).
Visual learning of Geometry
From the basic principles, geometry and its applications have developed a lot. Now, it has a vast area of application. To teach ideas of geometry, advanced study tools are necessary. It is almost impossible to learn 2D and 3D concepts of geometry without proper demonstrations. “The highlights, interlaced with interactive demonstrations, are intuitively developed. By learning to recognize patterns and powerful knowledge discovery process evolved” (Inselberg, n.d., para.1).
In order to learn different patterns, influential knowledge is required. It requires the help of geometrical concept. For example, recognition of M-dimensional objects form (M-1) requires lots to understand. For representing points in the plane, it is necessary that one should have knowledge regarding indices. This requires influential geometrical algorithm. That is, in order to make these algorithms application, knowledge is prerequisite. “Applications of parallel coordinates include collision avoidance and conflict resolution algorithms for air traffic control (3 USA patents), computer vision (USA patent), data mining (USA patent) for data exploration and automatic classification, optimization, decision support and process control” (Inselberg, n.d., para.3).
Conclusion
Geometry has got important role in life of the people, especially students. Geometry is considered to be important part of real life. Since “world is built of shape and space, and geometry is its mathematics” (Shape and space in geometry, 2010, para.5). Geometry is very helpful for the students in order to solve many problems. With the help of geometry, many students are presently solving many problems. This helps them to understand more. Finally, many people in the world are very well in thinking visually. In order to achieve this, geometry is considered to be doorway to achieve all the results. Students who are developing strong concept or intellect in the language of geometry can always excel in advanced topics related to mathematics. Thus, geometry is very important.
Reference
10 shocking reasons why geometry is important in your life. (n.d.). Math Worksheet Center. 2010. Web.
Dindyal, J. (n.d.). Algebraic thinking in geometry at high school level: Students’ use of variables an unknowns. Google docs. 2010. Web.
Finkbeiner, D.T. (1995). Recent publications. Mathematical Association of America, p.54. Web.
Geometry. (n.d.). Much More Math. 2010. Web.
Hickman, S. (2010). What types of jobs use geometry? eHow. Web.
How geometry is used in construction. (2010). Peerpapers.com. Web.
Inselberg, A. (n.d.). Parallel coordinates: Visual multidimensional geometry and its applications. 2010. Web.
Jordan, M. (n.d.). Why homework is important? Much More Math. 2010. Web.
Rawles, B.A. (2009). The geometry code: Symbolic wisdom of natural laws within us. Elysian Publishing. Web.
Shape and space in geometry. (2010). Annenberg Media. Web.
Mathematics is a very important subject because we use it in our day to day lives. Regardless of that, many learners express it as one of the most difficult subjects and that explains why many educators have been experiencing poor performance in this subject. This could be because most learners did not have a good foundation during their initial stages.
In mathematics, geometry is one of the most difficult subjects that pose many challenges to children. Children need to understand shapes, sizes, figures, and figures so as to appreciate geometry. This calls for proper foundation in geometrical concepts, both in schools and homes. Therefore, this paper will shed light on how educators can teach mathematics to children efficiently, particularly learning geometry.
According to Rich and Thomas (2008), the process of learning mathematics commences early enough even before the child reaches the age of going to school. But this study progresses automatically as the child gets acquainted to his or her surroundings. For instance, when you bring two toys to three children they will tell you that they are not enough and yet they do not know anything about numbers.
This is because they expect each one of them to have a toy. When a child is being introduced to mathematics, the teacher should start on a gradual pace by ensuring that the child first learns the basics. For instance you can never teach children how to add numbers when you have not taught them about numbers. This means that the basic lessons should come first.
Children gain knowledge through observation. Therefore, it would be important for the teacher to attract the attention of the child when he/she is demonstrating how the calculations are done. This can be achieved by asking questions at random to ensure that the children’s mind is glued to what is going on in the classroom. Moreover, asking questions helps the teacher to gauge the understanding of the learners (Clements, 2006).
If the teacher feels that a particular topic in mathematics was not well understood according to the performance of children in that topic, he/she should consider repeating that topic by using different approaches. Some of the methods that enhance understanding include selecting learners who understand the topic and have them demonstrate in front of the classroom how they were able to solve the sums.
The teacher should be present to make corrections where necessary. Above all, the teacher should be very patient when teaching children because their thinking capacity is still low and should consider asking questions about the things that were taught in the previous day before moving to another topic. This will help the teacher to identify the areas that need special attention.
Sarama and Clements (2006) explain that the teacher should pay special attention to all children without being limited to fast learners.
Besides, when the teacher does not engage children in his/her discussions, the children’s minds are most likely to be carried by other thoughts such as how they will watch the next cartoon episode. Moreover, listening in itself is a difficult task and that is why learners doze in class. This can be avoided by asking questions and also telling stories that relate to the topic being studied.
Mathematics is a very demanding subject hence the teacher should teach it when the kids are still fresh especially in the morning hours because in the afternoons the children are most likely to be exhausted. This is due to the fact that the time they spend on other subjects and as well as playing their games hence their level of concentration may decline.
Most teachers think that the best way to teach children mathematics is by giving them lengthy homework. This is very wrong because they may complete the assignments and yet they do not understand the concepts involved. In mathematics, the formula is the most vital element because unless the learner understands it their can be no answer to any mathematical problem.
It would be better if the child does a few sums that he/she understands than attempting a bulk of math that he/she does not understand. In such a case, the child will tackle the questions just to please the teacher and this may drive the child towards copying from peers which could continue to affect the child later in academic life.
The teacher should develop a habit of identifying slow learners in the classroom and keep an eye on their progress (Deiner, 2009).
Mathematics, especially geometry is best learnt through frequent exercises. This means that the child can be scheduled to solve four to five mathematical problems in a day. This goes a long way in preventing the situation where the child’s mind is congested with lots of formulas that the child can hardly remember.
When children are being introduced to geometry, it is important to teach them first about the geometrical apparatus such as the divider and the protractor so that when they come across a geometrical set they know how to use every tool including the compass. In addition, the children should be taught about the various geometrical shapes such as the triangles and rectangles among many other shapes.
Brumbaugh, Ortiz and Grasham (2006) state that while teaching a tough topic like geometry the teacher should integrate the parents and guardians to ensure that even after the child is out of school the parents and guardians will continue teaching the same topic to the child indirectly.
The parent can make the child understand the topic better by making them apply the geometrical skills in their plays and with the things that they interact with the most. For instance, the parent can ask the child to measure the width and length of the television set.
Parents can integrate geometry into the games children play. This includes making the child ride the bicycle in circles. The child can also be asked to measure the distance covered while riding the bicycle within the home compound.
Besides, the parent can ask the child to identify different shapes in the television programs the child watches. In addition, the parent can make snacks in different shapes to help the child understand the shapes better.
Deiner (2009) outlines that in geometry, the child’s understanding can be enhanced by displaying the various shapes and sizes in different pleasant colors. Besides, the teacher can also ask questions to the kid and provide assistance if the child gets stuck by giving a few hints towards the answer.
When the child works out a problem in the wrong way, the teacher should never give vague conclusions such as the answer was wrong or right but should rather elaborate the answer and help the child discover where he/she went wrong. This will make the child cautious about making the same mistake compared to when the teacher gives a vague remark.
Depending on the age of the child, the teacher can also employ arithmetic story books. This is in a bid to make the topic more interesting. The teacher needs to conduct assessment tests after covering a few areas of geometry.
The learners who achieve the highest marks should be rewarded with small gifts like cookies. Even without tests the teacher can motivate the children by requesting them to clap their hands for those that answer questions correctly.
Furthermore, children can be organized into small groups and then assigned problems to solve individually. In such case, the teacher should dig deeper into the child’s understanding by seeking to find out how the child at his answer.
This is accomplished by asking the child to explain why he gave a particular answer. Both the teacher and the parent need to be friendly to the child because if they are hostile or give lecture like remarks when the child makes a mistake it may demoralize the child.
The teacher can put on a warm smile in the classroom while the parent can offer a bar of chocolate during home based learning sessions. Both educators should also use a polite tone while speaking to the child. This also includes correctly choosing the words to use. The child should be made to identify the objects in his surroundings that are in the shapes taught in geometry class.
This can be items like plates, cups and beds among others. The parent should constantly remind the child about geometry by asking questions frequently such as when the child holds an item that has a geometrical shape (Garfias, 2011).
During class discussions every child should be allowed to express his views because that way the children will learn something from each other.
Besides, sharing their thoughts will provide a room for correction and thus build the child’s confidence while tackling such questions because he will remember what they learnt as a group. In some cases the children can be asked to write short essays about the topic. This practice aims at displaying their level of knowledge in the topic.
Harris and Turkington (2000) explain that practical exercises are also crucial in geometry because they enable children to demonstrate their skills. Such exercises can be carried out in a different location apart from the classroom such as in the play ground because they require more space for the shapes to be laid out.
The teacher can issue materials like blocks and porters mud and ask the kids to make the shapes they have learnt in class. Note that in this case there are no books to refer to.
In conclusion, geometry and mathematics in general should be made to look like a hobby for kids. If every child is provided with the appropriate guidance in understanding mathematics, the number of poor grades in science subjects that are reported in institutions of higher learning would diminish gradually because every learner would have changed his/her attitude.
Therefore, it is the duty of teachers and parents to assist children in learning mathematics.
References
Brumbaugh, K.D., Ortiz, E., & Gresham, G. (2006). Teaching Middle School Mathematics. New York: Routledge.
Clements, D. (2006). ”Ready for Geometry! From an Early Age, Children make Sense of the Shapes they see in the World around Them”. International Journal of Mathematical Education, Science and Technology. 2: 5-6.
Deiner, L.P. (2009). Inclusive Childhood Education: Development, Resources and Practice. New Delhi: Cengage Learning.
Garfias, L.E. (2011). ”Literal Math for Little Minds”. Whatever State I Am. Web.
Harris, J. & Turkington, C. (2000).Get ready! For Standardized Tests: Grade 2. New York: McGraw-Hill.
Rich, B. & Thomas, C. (2008). Schaum’s Outline of Geometry. New York: McGraw-Hill.
Sarama, J. & Clements, D.H. (2006). ”Early Math: Introducing Geometry to Young Children”. Scholastic. Web.
School geometry course has always been and remains one of the problematic “points” of teaching mathematics methods. At different times, various opinions were expressed about geometry and its place in the school system. Undoubtedly, the development of logic and intuition, which students observe in geometry, makes this discipline unique and necessary to study. One of the main goals of studying geometry in school is knowledge. However, it should be recognized that this goal concerning geometry is secondary since most school geometric knowledge is not in demand either in the practical life of a person or in scientific activity. The primary goal of studying geometry is that science is considered a phenomenon of universal human perception. Some theorems of geometry are among the most ancient monuments of world culture.
The task of updating the school geometry course is to make it modern and enjoyable, taking into account the inclinations and abilities of each student based on the level of geometric education achieved by the national school. In this case, the study of geometry depends on which textbook is used to teach geometry. A geometry manual should not be limited to building a geometric theory. The authors of all school primers try to make the presentation of the material as evidence-based as possible.
When teaching geometry in high school, it is necessary to pay attention to the formation of basic knowledge of the solid geometry course. At the same time, it is essential to find an opportunity to restore the basic understanding of the planimetry course. When studying geometry, it is crucial to increase the clarity of teaching, pay more attention to the issues of depicting geometric shapes, the formation of constructive skills and abilities, and apply geometric knowledge to solving practical problems.
Issues to Work on in the Course
It is possible to highlight the main problem that students face when studying geometry. In the study of geometry, the theory is critical. However, knowing theory alone is not enough to fully immerse oneself in a geometry course (Chan & Clarke, 2017). Often students, without hesitation, memorize the formulation of a theorem and its proof, but at the same time, they find it challenging to apply it. The inability to build a drawing, since it is a well-constructed drawing, is the key to success in solving a problem.
According to their drawing, schoolchildren try to make assumptions about any figure properties that are not specified in the assignment. For example, they build an isosceles triangle and start the solution, beginning from its properties, although there is no such condition in the task. Some schoolchildren cannot make a chain of logical reasoning that will lead to the solution of the study. It is also necessary to consider the peculiarities of the psychological development of schoolchildren of this age. Now, more than ever, an alternative school needs a balanced, well-thought-out system of geometric education.
The problem with geometry in high school is that students do poorly in readiness for Unit 1. Such conclusions were made when evaluating students according to the preliminary test. The data was collected at the beginning of the Modular Lesson on September 8, 2021. The problem with geometry test results occurs at Annapolis Road Academy. Traditional assessment tests do not provide sufficient information about the student’s inner abilities. They emphasize accurate information, including those procedures, and review skills that do not have an in-depth discussion.
An Expected Result After Solving the Problem
It is necessary to turn the class into a “math community” and not a gathering of people. The correct solution will encourage students to rely on their logic and mathematics since the training does not provide a straightforward solution for each task. It is essential to allow students to use their judgments and mathematical reasoning in place of memorization procedures. Moreover, it will enable students to reflect and be creative instead of asking only for the “correct” answer. Combining different aspects and rules with many mathematical positions is crucial instead of supporting the view that science is a system of separate concepts and skills.
Moreover, when using the assessment method exclusively, there is a tendency to distort the results to favor a specific group of students. At the same time, testing does not provide an objective assessment of the abilities of others. Practical testing aimed at assessing the level of knowledge of a large group of students requires a large number of documented technologies for collecting information on competency indicators. On the other hand, knowledge assessment is an integral part of the school curriculum. Simultaneously, learning is becoming more open and sensitive to distinctive features. It reflects any broad and accurate understanding that students have.
The model program for improving the quality of education is based on adequate consulting support. It aims to stimulate the professional growth of management and teaching staff in schools with low learning outcomes. The essential condition for improving the quality of education in geometry classes in Alternative schools with expected learning outcomes is developing a multi-level system of comprehensive support.
Moreover, the model program for improving the quality of education is based on interschool partnerships and networking of schools with different levels of quality learning outcomes. Its goal is to provide conditions for achieving comparable educational results by educational organizations in the region, primary schools with low learning outcomes, based on interschool partnerships and network interaction of schools with different levels of quality of learning outcomes. Teachers with a high level of professional competence can lead author’s schools and creative laboratories. Educators who experience difficulties organizing the educational process and who have a low level of special training will satisfy their professional needs by becoming listeners of an inter-municipal counseling center.
Knowledge assessment is an integral part of the learning process and pedagogy; systematic training is impossible without assessment. Nowadays, attention has begun to be paid not only to assessments focused on the final product but also to alternative assessments, which focus on the learning process (Yeo, 2017). The presentation of such authentic assignments as real-life situations increases the interest of students. The intellectual challenge provides competent data on individual learning and allows you to test various skills, including those necessary for a fulfilling life in modern society. They represent an antithesis to the prevailing assessment methods, which were considered insufficient, especially by professionals.
Traditional grading can be seen as the result of a “culture of testing,” while alternative grading can be considered a “culture of evaluation.” The classical assessment method is a traditional exam, which does not adapt to teaching methods (Ilany & Shmueli, 2021). Since the educational process involves attending full-time lectures, the exam aims to assess the material that students have memorized. However, the relationship between classroom learning and grade is not considered.
Special mention should be made of the relationship between geometry and the computer as a tool to increase the positive impact on the study of geometry. A laptop is a handy tool in geometric research. With its help, students can experimentally discover new interesting geometric facts. The person remains the most critical role in proving these facts. At the same time, from the point of view of mathematics, schoolchildren and the solid and weak technicians and humanities can be included in geometric activities using computers. The primary science, which is geometry, received a new impetus for development as an educational subject and as a science, thanks to the most modern computer technologies.
Methodology
After taking measures to improve the teaching and teaching of geometry, research improvements in understanding the subject should be carried out, and statistics should be collected. This research is to be conducted among 150 students in grades 10, 11, and 12 during their high school geometry courses, and the application of the study must take place in three phases. The research is carried out in the Alternative School and asks not only students but also teachers.
The first part of the study consists of three questions on geometry, regarded as a straightforward question, a simple question, and a difficult question. Students are given three different time frames to answer each question: easy, too easy, and challenging. Students should solve a “too easy” question in 4 minutes, an “easy” question in 6 minutes, and a “hard” question in 7 minutes, and they should suggest different ways of solving each question as much as possible. Each question in this app is designed so that students can imagine other ways of solving the questions.
The first question belongs to the category of simple questions and consists in finding the angle on a geometric figure. The task for students is also to find the answer in as many ways as possible. In the second stage of the study, a survey of 6 questions was applied. This review asked to explore their views on mathematics and mathematical issues and the various ways they used to address these issues. Students were asked to answer the questions by choosing the following appropriate options: 1: strongly agree, 2: agree, 3: disagree: 4: strongly disagree.
The last stage consists of a test of 20 questions, which are given 20 minutes. The test should be to identify students’ ability to solve problems on spatial perception. Students with a spatial ability score above 11 are coded as students with high spatial knowledge. In contrast, students with a spatial ability score equal to or below 11 are coded as students with low spatial ability.
Statistics are collected based on calculating student grades and giving out a total score for each quality. The study results should show the level of improvement that students and teachers demonstrated in teaching and mastering geometry.
Beneficiaries from Solving the Problem
It will benefit geometry students if problems are identified, and teachers know how to use the data, then they can help achieve the goal of improving student achievement. For teachers, the knowledge gained from the data analysis will be used to plan and deliver training. The teacher includes the standard in lesson plans and has proof of student progress in student work. The school’s goals are to include a commitment that promotes the following: a structured and predictable learning environment that minimizes unnecessary trauma; focuses on building positive and supportive relationships between students, teachers, and staff. It should take a balanced and refreshing approach to conflict and conflict resolution through skills development.
References
Chan, M. C. E., & Clarke, D. (2017). Structured Affordances in the Use of Open-Ended Tasks to Facilitate Collaborative Problem Solving. ZDM—Mathematics Education, 49, 951-963.
Ilany, B. & Shmueli, N. (2021). “Three-Stage” alternative assessment. Creative Education, 12(7), 26-44.
Yeo, J. B. (2017). Development of a framework to characterize the openness of mathematical tasks. International Journal of Science and Mathematics Education, 15, 175-191.
In modern society, geometry in architecture relates shapes with mathematical formulae in the realm of numbers and equations based on the coordinate number system. Typically, that is derived from the ancient view of geometry. However, the agency of geometry is based on drawing where the architect communicates meaning and ideas based on a set of conventions and views using lines that are mathematically connected to produce a coherent design to construct the object to completion (King, 2; Putz, 4). Thus, drawing relies on the agency of geometry in architecture, making both drawing and geometry complimentary.
One must exist in the realm of the other. With that brief argument in mind, the agency of geometry has been in existence in the architectural discipline with time and the complementary nature and significance of drawing and geometry as an agency in architecture bearing weight based on different architects. However, geometry as an agency and its significance have been a point of contention in the architectural world in the past centuries with different architects providing different views of the significance of geometry in their works.
A typical example is a difference existing between the perceptions of the significance of geometry in architecture in the works of Borromini’s, S. Carlo alle Quattro Fontane, Rome (1637-41) compared with that of Adam’s, Syon House, Middlesex (1761). Thus, there is a compelling urge to conduct an in-depth analysis of the views presented by different authors on the perceptions of the significance of the agency of geometry in both cases mentioned above (King, 2).
To search into the significance of geometry based on the perceptions of the architecture of the past centuries and their perceptions about the agency of geometry, it is crucial to briefly discuss the importance of geometry as perceived in the past centuries. The importance of geometry emerged in the 17th century when a church, despite the scarcity of financial resources and a tiny site, was erected at the four fountains intersection in Rome in 1643 based on a design by Francesco Borromini (King, 2). The design was a demonstration of the significance of geometry which showed how the design evolved from an elongated cross to an oval, then an octagon.
Typically, that design was based on an underlying geometrical structure that had different shapes amalgamated into the single geometrical shape, the design that was translated into the final building. Though it has been contended that these shapes had been redrawn in the successive centuries, other researchers contend that position by asserting the significance of geometry in the design and construction of the church in 1634 (Mostafavi, 3).
The geometry of the church had an underlying mathematical relationship in which different shapes were integrated into a single shape. These shapes included two triangles sharing a common base with perpendicular erections projected from their sides, and an inscription of two tangential circles that formed the point of focus that further yielded an oval inscription (Mostafavi, 3).
In addition to that, the geometrical shapes were mathematically related with a double rail rectangle that was tangential to the oval created in the short segment, and chapels that had a semicircular orientation along the major axis with four columns (Mostafavi, 3). The entire geometrical structure was reduced to an octagon by champers introduced into the corners of the rectangles leading to the completion of the design and the final structure of the chapel.
Typically, the importance of geometry provides the basis for argument on the significance of geometry in the cases of Borromini’s, S. Carlo alle Quattro Fontane, Rome (1637-41) and that of with Adam’s, Syon House, Middlesex (1761).thus, to provide a detailed view leading to the distinction between the perceptions of the importance of the agency of geometry in both cases, it is crucial to begin by examining the perceptions developed by Evans (Mostafavi, 3).
In both cases mentioned above, the differences in the agency of geometry are crystallized by discussing both cases about the significance of the agency of geometry from the perspective of Adam’s, Syon House, Middlesex, (1761) and that of Borromini’s, S. Carlo alle Quattro Fontane, Rome (1637-41) (Mostafavi, 3).
One of the writers who have discussed in detail the perspectives of geometry about Adam’s, Syon House, Middlesex, (1761) is Robin Evans. Robin Evans begins by viewing ordinary things as containing mysteries that are resolved as one makes an inquiry from the surface of things into the inner nature of things. While Borromini’s, S. Carlo alle Quattro Fontane, Rome (1637-41) begins their observations by seeing objects architectural structures with mathematical relationships, Adam’s, Syon House, Middlesex, (1761) views any object to consist of doors and windows that are culturally oriented and based on domestic architecture.
Typically, in their argument, information unfolds that shows Adam’s, Syon House, Middlesex, (1761) views to be more philosophical than mathematical in their arguments. One of the solid foundations of their argument is based on the geometry of the eye view. Typically, the argument shows that geometry is deeply founded in drawings that stimulate the imagination. However, these assumptions are confronted by the need to distinguish between architectural drawings and the artist’s use of drawings to stimulate imagination (Mostafavi, 3). However, it is crucial to distinguish the work of art and architectural drawings in the quest to discover the underlying evidence of the significance of geometry in architecture based on both views.
The distinction between architecture and art is based on the conclusion that the artist draws to communicate an idea in the form of a sculpture while on the other hand, an architectural drawing results in translating an idea in the form of a design into a building(Mostafavi, 3). That concludes the fact that the resulting design is based on the underlying geometrical principles. Typically, from the perspectives of Adam’s, Syon House, Middlesex, (1761), geometry provides the basis for translating one form to another without alterations. It is significantly important at this point since it provides the basis for designing anything that can be translated into a building.
Typically, “anything” can exist in the mind from which the idea is translated into a design and later into a building through geometrical means. These designs that are translated into buildings are based on the analogy of creating empty spaces in drawings. Typically, the drawing is presented using colors and other forms to communicate meaning by creating geometrical shapes that convey meaning to the intended audience. However, it is worth noting that Adam’s, Syon House, Middlesex, (1761) views on the significance of geometry draws on philosophical assumptions based on the view of the eye and the imagination of the mind (Mostafavi, 3).
Geometry, from Adam’s, Syon House, Middlesex, (1761) view indicates that geometrical shapes can be created using lines and the architect’s mind. However, at this point, Adam’s, Syon House, Middlesex, (1761) does not provide a clear view of the relationship between the geometry of the shapes created and their mathematical relationships. However, their argument draws heavily from the works of art, and the approach used to maneuver shapes making the projections tangible. Typically, it draws on the geometry of orthographic projections (Mostafavi, 3).
Orthographic projections provide a means for fitting geometrical shapes into others and providing the final object which in this case is a building. The geometry discussed here Adam’s, Syon House, Middlesex, (1761) is based on practical rather than a translation from drawings based on lines into the resulting object. That is demonstrated in the drawings of Palladio’s sketch that demonstrates the relationship between geometrical shapes and the close alignment between a drawing and a building is clear. Typically, the significance of geometry about the chapel mentioned above is based on orthographic projections, where one point of a drawing leads to the creation of another point in a drawing (Mostafavi, 3).
A typical example that discusses the perspectives of Adam’s, Syon House, Middlesex, (1761) based on orthographic projections shows how patterns emerge and are created as a result of other patterns. Typically, the orthographic projections can be in any angle of protection, either first or third angle projections. In either of the projections, one form of a drawing leads to the creation of diverse views of other views. In addition to that, the significance of geometry, in this case, is based on translating reality from the virtual world (Mostafavi, 3).
The following is a descriptive example of orthographic projections based on underlying principles of geometry used in the construction of a dome. According to the underlying geometrical principles is a dome with a floor consisting of curved ribs form the focus at an oculus ring.
Example of the projections of a dome.
It is possible to realize that the drawings are orthographic projections of a dome with the projection lines based on the geometrical principles that underlie the construction. Typically, the curvatures are not allocated names but are produced based on their points of intersections with other points in the dome. That results in several drawings being made from an original object that had its origin in the work of the imagination of the mind. Typically, the geometrical principles that, lead to the final project or product are based on orthographic projections. One can argue on the significance of orthographic projections, the imagination of the mind, the significance of a drawing, and their link to the geometry that translates a drawing into the final building (Mostafavi, 3).
Typically, one can see that parallel projections are produced on a plane that is circular onto a hemisphere that is further transformed into another figure with a requisite number of projections. Thus, formal circles lead to the generation of several lines and intersections leading to different views of the same design. The actual plan based on the above is the annular shape. Though the resulting figures look like circles or semi-circles, the geometry of origin is much more expendable and does not look like the final product which is a beautiful dome. However, it is crucial to work based on acquired technical drawing skills to produce such a design with the requisite accuracy for transformation into an actual design and gradually a building (Mostafavi, 3).
In theory, orthographic projections, according to Adam’s, Syon House, Middlesex, (1761) views are among the techniques used in the architectural world to provide and develop designs that translate into buildings (Evans, 25). Typically, the argument has its basis on projections, a radical departure from essentialism, a cultural perspective of architecture. In essence, one can see the underlying principles of orthographic projections to be based on the translations and transitions. Typically, translations provide the basis for transforming a drawing produced based on orthographic geometrical principles into a complete design and gradually a building.
It is worth, therefore, noting an additional view in Adam’s, Syon House, Middlesex, (1761) of the significance of geometry based on orthographic projections based on translations from one form into another (Michael, 1). However, in a nutshell, the translations and the projections are based on geometric patterns produced with such precision that little or no loss of the original object occurs in the transformation of the transition stage (Evans, 30). Having evaluated Adam’s, Syon House, Middlesex, (1761) views on the significance of geometry in architecture, it is worth examining the significance of geometry about the perceptions in Borromini’s, S. Carlo alle Quattro Fontane, Rome (1637-41) (Michael, 2).
In Borromini’s, S. Carlo alle Quattro Fontane, Rome (1637-41) views, the significance of geometry is based on another perspective and seems to show some radical departure from that provided by Adam’s, Syon House, Middlesex, (1761) views. In Borromini’s, S. Carlo alle Quattro Fontane, Rome (1637-41), geometry is the underlying rationale for a shape that is produced by an ordered sequence of items arranged in an orderly manner to produce the final design and building. Typically, the sequence consists of translations that occur in a specified ratio (Michael, 4). The underlying transformation is executed using lines as the connecting points for the transformations to occur.
One could readily see these lines as orthographic projections, thus, annulling the previous basis of arguments that geometry is based on orthographic projections, which is not the case. In the Borromini’s, S. Carlo alle Quattro Fontane, Rome (1637-41) case, there is the mathematical relationship that the former seems not the be aware of since their views originate from the mind and tend to take an artistic form which, however, radically departs when considering the underlying objectives of producing an actual building rather than a work of art such as a sculpture. In the apparent departure from Adam’s, Syon House, Middlesex, (1761) and yet a demonstration of the significance of geometry as provided by Borromini’s, S. Carlo alle Quattro Fontane, Rome (1637-41) views, a plan or design begins from a specific point as the origin of the entire plan.
However, both views regard drawings as the basis of any design (Michael, 6). A typical example in the drawing is just to identify a point and set the drawing instrument, in this case, the compass, open it to the right width and taking a specific point as the center, draw the first arch. Then, the next step follows by making arcs that make several intersections and center points to allow one to draw other arcs by taking the points of intersections as the centers of other subsequent arcs (Evans, 45). These arcs intersect at various points which are then joined to produce the desired design. It is important to note the significance of the geometrical intricacies introduced into the design by making several arcs that result in a specific design. Of course, the design can be translated into the final product which in this case is the building (Michael, 7).
In the design, due consideration is taken when producing the central points of intersection that lead to the creation of the Inner and Outer lines of San Carlo (Evans, 25). Typically, a summary of the construction process of the design is based on the use of a scale along the horizontal axis, and the setting of the compass used in the drawing to draw the bingo and the vertical axis along which the inner and outer lines of San Carlo are drawn (Michael, 10). As mentioned elsewhere, the geometry relies on the use of mathematical ratios, and these measurements are related to the curvature of the lateral chapel. It is important to see that each of the intersections produced and the resulting shapes have strong relationships with geometry used and the imagination of the mind.
It is worth noting that in the process of developing the plan for the San Carlo, site characteristics were also constructed as previously mentioned that it was constructed on a restricted site with limited financial provisions (Evans, 25). Thus, the gradual development and result of the plan showed that the resulting plan fitted exactly based on the geometrical intricacies that factored the construction site and the precision with which the design fitted into the selected site. However, without making a detailed analysis and description of the design and construction of the San Carlo chapel, it is important to evaluate the significance of geometry in the above case (Michael, 15).
It is critical to note that the author had formal knowledge of the techniques articulated by scientists of the time such as Kepler in formulating the geometrical relationships consisting of different shapes. In this case, Borromini makes use of the complex relationship existing between geometrical shapes and their mathematical relationships. In addition to that, it is worth noting that Borromini had developed a close relationship between the oval shape and geometrical relationships, by integrating both in the design to produce a design that fitted precisely into the requisite construction site.
Typically, Borromini’s regard for the complex relationships that can be developed using different geometrical principles is exemplified in the rigor with which the drawings of the San Carlo are produced with the underlying geometrical principles used to attain high precision. In addition to that, the San Carlo drawings are based on the use of lines with mathematical significance imbued in the geometrical principles used in the design of the building. However, it is crucial to note that both authors provide a sound basis for modern architecture and one is led to the conclusion that modern architecture can richly benefit from the experiences and perspectives of the significance of geometry in architecture when both are integrated.
Works Cited
Evans, Robin “The Developed Surface: An Enquiry into the Brief Life of an Eighteenth-Century Drawing Technique”, Translations from Drawing to Building and other Essays, London: Architectural Association, 1997. Print.
King, David. The Complete Works of Robert and James Adam, Oxford: Butterworth, 1991.
Michael, Hill, “The Curvature of the Lateral Chapels in San Carlo alle Quattro Fontane”, Julia Gatley (ed) Cultural Crossroads, SAHANZ: Auckland, 2009.
Mostafavi, Mohsen. Paradoxes of the Ordinary. Web.