-
-
--------------------- 2625 - EULER’S FORMULA - Topology for 6th Graders?
-
- Most everyone knows that a golf ball has dimples because aerodynamically the dimples cut down the amount of drag and air friction allowing the ball to travel faster and further. Look closely, the most efficient golf ball flyer is constructed from 232 indented polygon faces.
-
- But, look even more closely and you will see that they are not all hexagons. There are 220 hexagons and 12 pentagons. Why, because it is impossible to make a polyhedron of regular polygon faces for more than 5 solid objects.
-
- This fact was discovered in the year 1750. This review will define what a polyhedron is and the math that prevents the golf balls and soccer balls from being made of all hexagons.
-
- A polygon is a closed 2-dimensional figure having 3 or more straight line sides. A “regular” polygon has all sides that are equal and as many angles that are also equal. You know their names: triangles, squares, pentagons, hexagons, ……..
-
- A polyhedron is a 3-dimensional solid object or surface that is made up of these polygons as faces. A pyramid is made up of 4 triangles and on square at the base. A “regular” polyhedron is made up of all regular polygons. A pyramid, therefore, is an irregular polyhedron. Remember, there are only 5 regular polyhedrons. Here are their names:
-
---------------- Tetrahedron - has 4 triangular polygon faces. It is a type of pyramid with a triangle at the base.
-
---------------- Cube - has 6 square faces.
-
----------------- Octahedron - has 8 triangular faces. It is two tetrahedrons connected together at the base.
-
---------------- Dodecahedron has 12 pentagon faces. This one looks similar to a soccer ball, but, we will learn later that a real soccer ball has 12 pentagons and 20 hexagon faces. A real soccer ball is an irregular polyhedron.
-
----------------- Icosahedrons have 20 triangular faces.
-
----------------- That’s it, mathematically there can be no more regular polyhedrons.
-
- This mathematical rule was discovered and proven by Leonhard Euler in 1750. He was born in Switzerland in 1707. His name is pronounced “oiler”. And, he is one of the greatest mathematicians that has ever lived.
-
- It all started with this simple formula, called Euler’s formula. The polygon faces all have edges where the polygons come together. Also, they all have points, or solid angles called vertices, where the edges come together. For all the polyhedrons count the number of vertices, edges, and faces. Euler’s formula says:
-
----------------------- vertices - edges + faces = 2
-
----------------------------- V - E - F = 2
-
- This formula is so simple yet so powerful you will not believe. Always equals 2. It introduces a whole branch of mathematics called “topology“. Most all objects can be defined with points, lines and surfaces. To begin, let’s first apply the formula to our 5 regular polyhedrons:
-
----------------- Vertices - Edges + Faces = 2
-
----------------Tetrahedron: 4 - 6 + 4 = 2
-
---------------- Cube: 8 -12 + 6 = 2
-
----------------Octahedron: 6 - 12 + 8 = 2
-
----------------Ddecahedron: 20 - 30 + 12 = 2
-
----------------Icosahedron: 12 - 30 + 20 = 2
-
- Now let’s apply Euler’s formula to a soccer ball. A soccer ball has 32 faces, 12 are pentagons, 20 are hexagons. There are 90 edges and 60 vertices.
-
---------------- V - E + F = 2
-
---------------- 60 - 90 + 32 = 2
-
- Let’s apply Euler’s formula to a sphere. Wait a minute. A sphere has no edges or vertices, it is one smooth surface, or face. Use your imagination. The Earth is a sphere, but, in your imagination you can slice it up with 12 longitudinal lines from pole to pole.
-
- Then, slice it horizontally east to west with 7 latitude lines running parallel to the equator. Those imaginary lines would divide up the Earth into 96 faces of polygons. There are 180 edges and 86 vertices:
-
---------------- V - E + F = 2
-
---------------- 86 - 180 + 96 = 2
-
- Euler’s formula applies to all types of solids and surfaces and it is not always equal to 2. A different equation will define a different type of solid. Let’s take a donut for example. This category of shape is called a “torus”, but the donut is more familiar. In this case:
-
---------------- V - E + F = 0
-
- To create the vertices, edges, and faces on the donut first cut it in half like a bagel. Then cut each half into 4 equal pieces. Resemble the donut and you have 8 faces of polygons, 16 edges, and 8 vertices:
-
---------------- V - E - F = 0
-
---------------- 8 - 16 + 8 = 0
-
- If a surface area has a Euler formula that is not equal to zero it is not a “torus“. If a surface area has a Euler formula that is not equal to two it can not be a “sphere“. There are many, many more formulas to describe all kinds of surfaces in the study of topology.
-
- Topology would define a donut and a coffee cup as having the same shape. If you are intrigued by this idea google Mobius Band and Klein Bottle to learn more.
-
- Math is fun. See other reviews available
-
- 2604 - Math for fun
-
- 2571 - Math was invented to solve problems.
-
- This Review 2571 lists 43 other reviews all about interesting math solutions.
-
- February 18, 2020 1033 2625
----------------------------------------------------------------------------------------
----- Comments appreciated and Pass it on to whomever is interested. ----
--- Some reviews are at: -------------- http://jdetrick.blogspot.com -----
-- email feedback, corrections, request for copies or Index of all reviews
--- to: ------ jamesdetrick@comcast.net ------ “Jim Detrick” -----------
- https://plus.google.com/u/0/ -- www.facebook.com -- www.twitter.com
--------------------- Tuesday, February 18, 2020 --------------------
-----------------------------------------------------------------------------------------
No comments:
Post a Comment