Maths Ia Break The Rules Of Conventional Geometry Wuxi No.1 High School Essay

1312 words - 6 pages

Break the rules of conventional geometry
Introduction
My doubt on the Euclidean geometry that the sum of angles in a triangle is 180°
started from my experience on one birthday party.Many balloons were decorated in
the room and there were some pictures on the balloons,I found that triangles on the
balloons were not the same as what I see in common—they are elliptic
triangles(see in figure1).Then a wonder created in my mind that whether the sum of
angles in those triangles is larger than usual.
Figure.1
The first step of breaking the conventional rule is to prove it.
Euclidean geometry
Euclidean geometry is a mathematical system attributed to the Alexandrian Greek
mathematician Euclid, which he described in his textbook on geometry: the
Elements. Euclid's method consists in assuming a small set of intuitively appealing
axioms, and deducing many other propositions (theorems) from these.
One of the important well-known results is the triangle angle sum—the sum of the
angles of a triangle is equal to a straight angle (180 degrees). This causes an
equilateral triangle to have 3 interior angles of 60 degrees. Also, it causes every
triangle to have at least 2 acute angles and up to 1 obtuse or right angle.
Proving Euclidean geometry
There is a given triangle:
Draw a line parallel to side BC of the triangle that passes through the vertex A.
Label the line as PQ.
∠PAB+∠BAC+∠CAQ=180º( Because PQ is a straight line which means ∠PAQ is
180º)
∠PAB=∠ABC(congruent as PQ//BC and AB transversal line)
∠CAQ=∠ACB(congruent as PQ//BC and AC transversal line)
so, in the triangle ABC,
∠BAC+∠ABC+∠ACB=180º
In the above situation, there is precondition that the triangle must follow Euclid’s
fifth postulate which “Through a point not on a line, there exists one and only one
line parallel to this line.”In other words, this rule is true in the plane, but in some
other types of surfaces, it may not be.
In mathematics, we divide various surfaces depending on Constant Gaussian
Curvature, some are surfaces of Constant Gaussian Curvature and others are not.
In common, surfaces of Constant Gaussian Curvature are three below:sphere,
pseudosphere, and Euclidean plane.
* Gaussian Curvature:Gaussian curvature is an intrinsic measure of curvature,
depending only on distances that are measured on the surface, not on the way it is
isometrically embedded in Euclidean space.
To explore the sum of angles in the triangle in the sphere,I designed an
experience .
Experiment
Figure 2
Sphere
Figure 3
Pseudosphere
Figure 4
Euclidean plane
the sum of angles in the triangle in sphere.
Step1
Prepare a ping-pong, a tennis ball and a basketball. (different sizes)
Step2
Measure the radius of three balls.
Use two pieces of board to fix the ball, and then mark the position of board,
measure the shortest distance between two marks(the diameter of the ball) and use
the radius formula
d=2r r=d/2
to calculate the radius of the ball.
Step3
Draw a triangle on the ball.
Learned from the three-dimensional...

Other Essays On Maths IA--Break the rules of conventional geometry - Wuxi No.1 High School - Essay

change school rules to make schoolo a better place - mariner high school - languge arts

513 words - 3 pages I learned a lot on the field trip to the federal reserve. I learned about how money is prossed. I learned that banks send a lot of money to the federal reserve so they can process it and make sure that is all clean money, and to make sure the money is real. Most of the banks around the atlanta area send there money to the federal reserve. After they finish processing the money they put the money on a yellow auto piloted robot that forklifts the

Identify Amir’s strongest emotions in Chapters 1-12 - Shanghai International high school, Literature class - Essay

1462 words - 6 pages Amy Yao EN201-1 Identify Amir’s strongest emotions in Chapters 1-12. Which emotions are persistent, and which change over time? Based on these emotions and how are presented, compare Amir with one other character from the novel and judge (a)which of these characters is more in touch with their true emotions and (b) which of them is more justified in feeling the way they do. “That was a long time ago, but it’s wrong what they say about the past

Homeostasis and how it effects the body - health science 1 concord high school - assignment

892 words - 4 pages -is-homeostasis/. YourDictionary. “Homeostasis Examples.” YourDictionary, 28 July 2016, examples.yourdictionary.com/homeostasis-examples.html. khan academy. “Homeostasis.” Khan Academy, Khan Academy, www.khanacademy.org/science/high-school-biology/hs-human-body-systems/hs-body-structure- and-homeostasis/a/homeostasis. BiologyWise Staff. “A Brief Introduction to the Levels of Organization of Living Things.” BiologyWise, BiologyWise, 6 Aug. 2017

Abraham Lincoln and the Struggle for Union and Emancipation - Bridgewater Raritan High School; APUSH 1 - DBQ

698 words - 3 pages President Abraham Lincoln faced an incredible challenge during his time as Commander-in-Chief of the United States: reuniting the Union. Lincoln’s sole purpose in fighting the Civil War was to mend the break between the North and the South. Fortunately, Lincoln realized he had to accommodate changes to the war plan if it helped him achieve his ultimate goal. On January 1, 1863, Lincoln issued the Emancipation Proclamation, establishing the

The sun also rises chapters 1-3 - Hillsboro high school American men - Assignment

480 words - 2 pages , and I always had a suspicion that perhaps…”( Hemmingway, 12) Jake feels threatened by a superior, composure and it makes him angry. Jake is also a hard worker 1. Discuss the themes of the text through the first 3 chapters. Drinking to fill in time is a big theme in this book. When anything remotely difficult happens, Jake drinks. Traveling for a purpose 2. Discuss/identify the setting of the first 2 chapters. The first two chapters were in the

Romeo and Juliet Essay on how they were so in love and that no one can interrupt their elationship. - Monsignor Paul Dwyer Catholic High School - Essay

908 words - 4 pages Jackson 1 Tuoyo Jackson Mrs. Prebble ENG1D1 November 24, 2016 Romeo and Juliet Essay Assignment “The fool doth think he is wise, but the wise man knows himself to be a fool.” “Love all, trust a few, do wrong to none.” In Shakespeare’s Romeo and Juliet, Romeo and Juliet are doomed as they had no way of preventing the inevitable due to the outside forces such as the secrecy, parental influence and the lack of responsibility of Friar Laurence and

Causes of the french revolution DBQ - haverhill high school - essay

1289 words - 6 pages poor little horse, yet they had to pay a tax of 42 pounds of wheat, and three chickens, to one noble and 168 pounds of oats, one chicken and 1 sou [small unit of money] to another...the taxes and laws are crushing us. This woman, at no great distance, might have been taken for sixty or seventy, her figure was so bent, and her face so wrinkled and hardened by labor, — but she said she was only twenty-eight.” Q5: List three observations this

The Evolution of Canadian Bears - Nepean High School/Biology - Essay

1167 words - 5 pages brown bear from North America to mate with the asiatic black bear of Asia, an example of genetic drift. The bridge was submerged approximately 14,000 years ago, resulting in zero to no gene flow between species, due to habitat isolation. In their own environments, bears have become structurally, behaviourally and genetically diverse from their relatives. Split from early dog-like creatures (canis) about 20-25 million years ago, bears are believed

The restricted freedom of women - Montgomery Blair High School - essay

714 words - 3 pages Becoming a Springbrook Blue Devils Cheerleader A varsity cheerleader must: 1. Complete all necessary paperwork on time. 2. Be academically eligible. 3. Must attend scheduled games and practices. 4. Be physically fit – you must have the ability to maintain a high level of endurance for an extended period of time. 5. Demonstrate ability to perform cheer skills such as motions, stunting, tumbling and some dance skill. 6. Display a positive attitude and

Theme of Savagery in Lord of the Flies - High School Essay - English Essay

876 words - 4 pages equal say and opinion. If they have the conch, no matter who they are or what age they are they will be given the chance to speak and will be listened to by the rest of the boys. The boys have created the island to be a democratic place which shows a civilized side to them as they try to mimic the homes they have just left. Contrasting with the symbol of the conch is the symbol of the beast which comes to be associated with Jack as by the end of

This an essay of the film Children of Men. - Mount Lawley Senior High School - Essay

927 words - 4 pages Practice Question – Children of Men Question 1: Discuss the way the exposition or the resolution of a text studied enables the audience to understand the ideas it conveys. The film, “Children of Men” directed by Alfronso Cuaron portrays the chaotic world in which people are dealing with the consequences of the infertility. The film upholds common ideologies and issues which is relevant to the present world and this becomes prevalent towards the

Similar Papers

About Maths How We Are Using Geometry In Real Life 55 School 9 1 Research Paper

1424 words - 6 pages  celestial sphere and describing the relationship between movements of celestial bodies. In the field of Physics, there is a deep link between pseudo-Riemannian geometry and general relativity · Geographic Information Systems 240 Geometry concepts are used in satellites in gps systems, it calculates the position of the satellite and location of gps gauged by the latitudes and longitudes. Maths thinking and reasoning begins for students long before it

Declaration Of Independence Federalist No. 44 Gregory Portland High School Essay

441 words - 2 pages Rena Ross K. Lovas GOVT 2305 Fall 2018 FEDERALIST No. 44 Utilizing a strict tone, James Madison asserts that the states shall not maintain a vast amount of power over Congress; the states must follow the rules; abide the rules; and shall not attempt to overthrow or disclaim the rules. “The Constitution and the laws of the United States...under the authority of the United States, shall be the supreme law of the land, and the judges in every state

Assignment 1 The Achievement Of Desire Great Bridge High School/ Ap Eng 12 Assignment

736 words - 3 pages 1 Gordnier All throughout my life I always felt like I was living in the shadow of my older brother. Because I’m the younger sibling, there was already a preconceived idea of who I was supposed to be and what I was supposed to accomplish based on his previous success. He was a straight A student in the top percentile of his class, charming his teachers with his natural ease in academics. He played soccer and was in the school band, showing off

Masque Of The Red Death Literary Analysis Next High School, English 1 Literary Analysis

460 words - 2 pages Bedley Bedley Taran Bedley Ms. Bright English 1, Period 3 October 3, 2017 Literary Analysis of the Masque Red Death In “The Masque of the Red Death,” Poe uses the clock, the final chamber, and the Red Death to express that death is unavoidable. The clock, an imposing figure in the western chamber, symbolizes the inevitability of death. Each hour, the clock sounds the hour with its powerful “and so peculiar note” (Poe) that both dancers and