Polyhedra

Prisms and Antiprisms

In geometry, a prism is a polyhedron with an n-sided polygonal base, another congruent parallel base (with the same rotational orientation), and n other faces (necessarily all parallelograms) joining corresponding sides of the two bases. Prisms are named for their base, so a prism with …Read the Rest

Truncated Platonics

2015-02-02
Truncated Platonics

There are several Archimedean solids that are formed by the truncation (cutting off) of each corner of a Platonic solid. These can be shown in successive truncations from one shape to its dual. Original Truncation Rectification Bitruncation Birectification (dual)   Tetrahedron   Truncated Tetrahedron   …Read the Rest

Icosahedron

2015-01-29
Icosahedron

The icosahedron has 12 equilateral triangles as faces. It can be split into 3 parts, a pentagonal anti-prism and two pentagonal pyramids. We will start by looking at a pentagonal pyramid. We have already done most of the work earlier. To find ∠ACB, $$\begin{align}\cos \angle …Read the Rest

Dodecahedron

2015-01-28
Dodecahedron

The dodecahedron is a regular polyhedron made of 12 regular pentagon faces. A D12 die is often used in certain role playing games. Since a dodecahedron has 12 sides, you can print out a calendar, one month per face, and have a conversation piece on …Read the Rest

SketchUp Platonics

2012-01-24
SketchUp Platonics

I’ve been playing with Google’s Sketchup for a week or so and I had the thought of doing some polyhedra. It is often hard to mentally visualize a 3D object when looking at a 2D image, so I figured these would help. Here is the …Read the Rest

Octahedron

2011-11-07
Octahedron

The octahedron is a Platonic solid, which means it is made of all regular polygons for each face, being eight equilateral triangles arranged four at each vertex. With edge length of S, the surface area would be that of 8 equilateral triangles, \(8\cdot S^2\cdot \frac{\sqrt3}{4}=2\cdot …Read the Rest

The Cube

2011-10-28
The Cube

The cube is probably the most recognized and best known of and 3D shape. Kids young enough not to even know how to talk will still know about cubes, they play with multicolored wooden blocks. Older kids may tackle the Rubik’s Cube puzzle or play …Read the Rest

An unequal pyramid

2011-10-11
An unequal pyramid

Last post, I worked on the vertex edge angles of a triangular pyramid that had all equal angles originating from the apex. Using the same camera tripod analogy, take one of the legs (line DA) of the tripod and slide it out further from the …Read the Rest

Tetrahedron

2011-10-10
Tetrahedron

I started off posting about the truncated icosahedron, despite being fairly complex compared to the platonic solids, like the tetrahedron. The regular tetrahedron is probably the simplest 3D shape, except for maybe the sphere. It is made of 4 equilateral triangles, forming a triangular pyramid. …Read the Rest

Finding the Angles of the Truncated Icosahedron

I’ve been interested in various geometric shapes for many years, from the simple cubes, prisms, and pyramids made in middle school, to the geodesic domes used to build houses. I have cut and glued numerous pieces of paper into just about all of the regular …Read the Rest