Yen's Mathematics Daily

a Johnson solid is a strictly convex polyhedron each face of which is a regular polygon. There is no requirement that each face must be the same polygon, or that the same polygons join around each vertex. An example of a Johnson solid is the square-based pyramid with equilateral sides (J1)

Although there is no obvious restriction that any given regular polygon cannot be a face of a Johnson solid, it turns out that the faces of Johnson solids which are not uniform (i.e., not a Platonic solid, Archimedean solid, uniform prism, or uniform antiprism) always have 3, 4, 5, 6, 8, or 10 sides.

In 1966, Norman Johnson published a list which included all 92 Johnson solids (excluding the 5 Platonic solids, the 13 Archimedean solids, the infinitely many uniform prisms, and the infinitely many uniform antiprisms), and gave them their names and numbers. He did not prove that there were only 92, but he did conjecture that there were no others. Victor Zalgaller in 1969 proved that Johnson's list was complete.

As in any strictly convex solid, at least three faces meet at every vertex, and the total of their angles is less than 360 degrees. Since a regular polygon has angles at least 60 degrees, it follows that at most five faces meet at any vertex. The pentagonal pyramid (J2) is an example that has a degree-5 vertex.

A database of solids and polyhedron vertex nets of these solids is maintained on the Sandia National Laboratories Netlib server (https://netlib.sandia.gov/polyhedra/), but a few errors exist in several entries. Corrected versions are implemented in the Wolfram Language via PolyhedronData.

The following list summarizes the names of the Johnson solids and gives their images and nets.

In geometry, a skew apeirohedron is an infinite skew polyhedron consisting of nonplanar faces or nonplanar vertex figures, allowing the figure to extend indefinitely without folding round to form a closed surface.

I built some of my polyhedron

I made some my sphere with inspiration from patterns found in Mobius transformation 

Here is GIF by hyperbolic-gifs.tumblr.com

Odd Third

Saw this fabulous fraction at the Futility Closet, with a link to Roger Nelson's triangular proof without words. When I was making sense of it myself, I thought about how the sum of the odds is a square number, so the first n odds 1, ..., 2n-1 sum to n^2, But then the first 2n odds sum to (2n)^2, so the 2nd n odds, 2n+1, ..., 4n-1 sum to 3n^2 and I had to make this visual.

That it goes back to Galileo is just gravy!

I built some of my polyhedra (Polyhedron, Uniform polyhedron compound)

Also, Object of my inspiration is here:

The book Polyhedron Models by Magnus Wenninger

Re-Tie