November 21, 2009
November 20, 2009
November 19, 2009
November 18, 2009
Geometry Notes — History — The Platonic Solids
The Platonic Solids
The Platonic solids are regular 3-D figures. A Platonic solid is a convex regular polyhedron. Regular means that all the edges are of equal length, all the angles of equal measure, and all faces are congruent shapes. They are unique in that the faces, edges and angles are all congruent. There are exactly five such figures. The name of each figure is derived from the number of its faces: 4, 6, 8, 12, and 20. Tetrahedron — 4 faces. Hexahedron or Cube — 6 faces. Octahedron — 8 faces. Dodecahedron — 12 faces. Icosahedron — 20 faces.
The following gives the name of the Platonic Solid along with the number of faces it has, the shape of each face (which is a regular polygon), the number of vertices, the number of edges, and the Platonic Solid’s Dual which is the Platonic Solid that can be inscribed within it by connecting the midpoints of the faces.
Platonic Solid Faces Shape of Faces Vertices Edges Dual
Tetrahedron 4 Triangle 4 6 Tetrahedron
Cube 6 Square 8 12 Octahedron
Octahedron 8 Triangle 6 12 Cube
Dodecahedron 12 Pentagon 20 30 Icosahedron
Icosahedron 20 Triangle 12 30 Dodecahedron
The ancient Greeks studied the Platonic solids extensively.
Some sources credit Pythagoras and the Pythagoreans with their discovery. Other evidence suggests that they may have only been familiar with the tetrahedron, cube, and dodecahedron, and that the discovery of the octahedron and icosahedron belong to Theaetetus, a contemporary of Plato. Theaetetus gave a mathematical description of all five and may have been responsible for the first known proof that there are no other convex regular polyhedra.
The Platonic solids feature prominently in the philosophy of Plato. Plato theorized the classical elements were constructed from the regular solids. Plato was mightily impressed by these five definite shapes that constitute the only perfectly symmetrical arrangements of a set of non-planar points in space, and later expounded a complete “theory of everything” (in a writing called Timaeus) based explicitly on these five solids. To achieve perfect symmetry between the vertices, it’s clear that each face of a regular polyhedron must be a regular polygon, and all the faces must be identical. Plato speculated that these five solids were the shapes of the fundamental components of the physical universe. Plato associated each of the four classical elements (earth, air, water, and fire) with a regular solid. Fire was associated with the tetrahedron, earth with the cube, air with the octahedron, and water with the icosahedron. The fifth Platonic solid, the dodecahedron, Plato associated with the universe. Aristotle added a fifth element, aithêr (“ether” in English) and postulated that the heavens were made of this element, but he had no interest in matching it with Plato’s fifth solid.
The aesthetic beauty and symmetry of the Platonic solids have made them a favorite subject of geometers for thousands of years. The Platonic solids have been known since ancient times. Ornamented models of them can be found among the carved stone balls created by the late neolithic people of Scotland at least 1000 years before Plato.
Now for a technical point. There is a formula that associates the number of faces, the number of vertices, and the number of edges of any convex polyhedron. The adjective convex is very important. This formula works for the perfect Platonic solids as well as any convex polyhedron. Cubes are nice regular hexahedrons. They get all the credit and praise. They are pretty. But a shoe box, for example, is also a hexahedron. The formula F – E + V = 2 holds true for all polyhedron. F is the number of faces. E is the number of edges, and V is the number of vertices.
We call this formula Euler’s formula, named after Leonhard Euler. Leonhard was responsible for much of early work to prove various theorems about polyhedra.
Tetrahedron (recall 4 vertices, 6 edges, 4 equilateral triangles as faces) 4 – 6 + 4 = 2
Hexahedron (recall 8 vertices, 12 edges, 6 squares as faces) 6 – 12 + 8 = 2
Here is a nice website. It shows the Platonic Solids in motion. Click and hold on one, move the mouse. You can even get them to roll around. Kinda cool.
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November 17, 2009
Geometry Notes — Kite
A kite is a member of the quadrilateral family, and while easy to understand visually, is a little tricky to define in precise mathematical terms.
A kite is a convex quadrilateral with two adjacent sides the same length and the other two adjacent sides the same length.
Things I know about kites…
- 1. 4 sides and 4 angles
2. the sum of the interior angle is 360 and the sum of the exterior angles is 360
- 3. Two pairs of adjacent sides of a kite are equal in length
- 4. One pair of opposite angles (the ones that are between the sides of unequal length) are congruent.
- 5. One diagonal bisects the other. They do not bisect each other, however.
- 6. Diagonals are perpendicular.
- 7. One diagonal divides a kite into two isosceles triangles; the other (the axis of symmetry) divides the kite into two congruent triangles.
A kite can become a rhombus. In the special case where all 4 sides are the same length, the kite satisfies the definition of a rhombus. If this happens, the figure doesn’t stop being a kite. But a better classification would be rhombus, which places it in the parallelogram family of quadrilaterals. And as you know, a rhombus can become a square if its interior angles are 90°.
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Dart – Concave quadrilateral with two sets of congruent adjacent sides
Beam me up Scotty.
Pretty stuff.
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Geometry Notes — Trapezoid, Isosceles Trapezoid — Sketchpad Slides
Trapezoid
Isosceles Trapezoid
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