Phi, the Golden Ratio

Phi \((\Phi, \phi)\) is a Greek letter that mathematicians have assigned to a specific ratio or proportion, called the golden ratio, that most people find to be attractive in art, architecture, and nature. The golden ratio is illustrated as \(\frac{A}{B}=\frac{A+B}{A}\equiv\phi\). The only positive solution is …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 from around the internet. Many of these formulas can be written in different ways, but I have simplified them as much as possible. Where, \(\phi=\frac{1+\sqrt{5}}{2}=1.6180339887498948482045868\ldots\), …Read the Rest

22 miles straight up in 90 seconds

I may not involve geodesic shapes, but sure is awesome. This is not a “because we can” moment, it is a “because nobody else can” moment. This is science at its finest. It hits Mach 3!!! Also see:

An unequal pyramid

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



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