Quite often, when doing calculations on polyhedra, you will find yourself with complex equations like $$\begin{equation}\tag{eq1}\label{eq1}\frac{20+7\sqrt3}{1+2\sqrt3}\end{equation}$$ Is the top evenly divisible by the bottom? I certainly can’t tell just by looking at them. There must be a method to factor the numerator. Let’s start with $$(A+B\sqrt{R})(C+D\sqrt{R})$$ multiplying the terms together gives $$AC+BRD+(BC+AD)\sqrt{R}$$ So, $$\frac{AC+BRD+(BC+AD)\sqrt{R}}{A+B\sqrt{R}} = C+D\sqrt{R}$$ Then we can substitute known values (A=1, B=2, R=3) into the formula. $$\frac{1C+2(3)D+(2C+1D)\sqrt{3}}{1+2\sqrt{3}} = **…Read the Rest**

### sqr(A+B*sqr(C))

Lately, I have been doing a lot of math involving square roots of numbers added to square roots, in the **…Read the Rest**

### SketchUp Platonics

I’ve been playing with Google’s Sketchup for a week or so and I had the thought of doing some polyhedra. **…Read the Rest**

### Octahedron

The octahedron is a Platonic solid, which means it is made of all regular polygons for each face, being eight **…Read the Rest**

### Law of Sines and Cosines

The law of cosines relates the sides and angles of a triangle. \(a^2=b^2+c^2-2bc\cdot \cos\alpha \\ b^2=a^2+c^2-2ac\cdot \cos\beta \\ c^2=a^2+b^2-2ab\cdot \cos\gamma\) **…Read the Rest**

The cube is probably the most recognized and best known of and 3D shape. Kids young enough not to even **…Read the Rest**

### Phi, the Golden Ratio

Phi \((\Phi, \phi)\) is a Greek letter that mathematicians have assigned to a specific ratio or proportion, called the golden **…Read the Rest**

### Table of exact trigonometric functions

Since I noticed I had to keep looking some of these up, I place them here, just for reference, gathered **…Read the Rest**

### 22 miles straight up in 90 seconds

I may not involve geodesic shapes, but sure is awesome. http://hackaday.com/2011/10/10/22-miles-straight-up-in-90-seconds/ This is not a “because we can” moment, it **…Read the Rest**

### An unequal pyramid

Last post, I worked on the vertex edge angles of a triangular pyramid that had all equal angles originating from **…Read the Rest**

### Tetrahedron

I started off posting about the truncated icosahedron, despite being fairly complex compared to the platonic solids, like the tetrahedron. **…Read the Rest**