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ADAM CLEMENTS SME 430 Discussion Post #6 2/23/2010

Complete the readings for this week and then post your reflections to the prompts below (15/opening)

1. What are the names of the Platonic Solids? Tetrahedron Hexahedron (cube) Octahedron Dodecahedron Icosahedrons

2. What are the properties of the Platonic Solids? Since the polyhedron is regular, the situation at any vertex is the same at any

other. Therefore, we only have to consider what happens at a typical vertex.

In order to make a peak, the sum of all the face angles at the vertex must be lessthan 360 degrees. (if they added up to exactly 360 degrees they would make a flat

surface.)

Since all the faces are congruent, the angle sum at a vertex must be divided upequally among them.

Tetrahedron 4 faces (triangles) Hexahedron (cube) 6 faces (squares) Octahedron 8 faces (triangles) Dodecahedron 12 faces (pentagons) Icosahedrons 20 faces (triangles)

3. Pick one Platonic Solid and show how it satisfies the properties. The cube has 6 square faces All vertexes are the same The sum of the face angles at the vertex add up to 270 degrees. (90 x 3 = 270) Each vertex is 90 degrees.

4. The Pythagoreans associated the Platonic solids with certain elements. What are theassociations they used?

Fire = tetrahedron Earth = Hexahedron (cube) Air = Octahedron Water = Icosahedrons The fifth element represented by the dodecahedron was supposed to be the

fundamental essence of the universe.

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Identify one pair and explain why they might have chosen to associate that element

with that solid?

Tetrahedrons are pointed like a flame thus this might explain why they chose toassociate them together.

5. What is the distinction between Platonic solids and Archimedean Solids? Platonic solids are all made up of the same shape. Archimedean solids can be made up of more than one different shape.

6. How many Archimedean Solids are there? Name one example. Archimedes discovered that there are exactly 13. A truncated icosahedrons made up of 20 hexagonal faces and 12 pentagonal ones.

This is known to many as a soccer ball.

7. Identify 2 Contributions from Egyptian Mathematics history.1. To multiply or divide, they used an ingenious method based on doubling.2. They knew how to computer or approximate the areas and volumes of several

geometric shapes, including circles, hemispheres, and cylinders.

8. Identify 2 Contributions from Babylonian Mathematics history.1. They use of more extensive tables of products, reciprocals, and conversion

coefficients.

2. They knew and applied the Pythagorean Theorem, and had formulas forapproximating areas of volumes of various commons shapes.

9. Identify 2 Contributions from Greek Mathematics history.1. By putting logical reasoning and proof at the center of the subject, they changed

forever what it means to do mathematics.

2. They were engaged in working out formal proofs of mathematical statements.They began to understand that in order to prove theorems, one must start with a

few unproved assumptions. These were taken for granted and considered as

truths. This is the oldest Green mathematical work that still survives intact.