# 120-cell explanation

The 120-cell is the four-dimensional equivalent of the three-dimensional regular dodecahedron.

Just as the three-dimensional dodecahedron is made up of 12 two-dimensional pentagonal faces, the four-dimensional 120-cell is made up of 120 three-dimensional dodecahedral "faces".

The models shown here are parts of a projection of the 120-cell into ordinary three-dimensional space.

The basic modules, tubes with triangular cross-sections, represent the edges of regular pentagons as well as of platonic dodecahedrons and of the 120-cell. The length and width of the tubes are determined by the width of the tape used to fold the models.

As is usual with origami, there is no glue used to connect the modules. Instead, they are “snapped” together by other modules. Therefore the name Snapology for this folding technique.

As all edges of the models are of constant length, the reduction of lengths that take place in the projection process (perspective foreshortening) cannot be reproduced exactly. So the models are restricted to parts of the projection by omitting those edges that are too much foreshortened by perspective. Smaller amounts of perspective foreshortening can be overcome by making use of folding tolerances and the resilience of the paper.

Sometimes the perspective foreshortening of a pentagon is overcome by using a five pronged star instead.

The series of models try to construct the 120-cell, starting with a single dodecahedron in the centre (3D-view) or “south pole” (4D-view). Consecutively further shells (3D-view) of dodecahedrons (or parts of them) are added. As further shells are added, inner shells that would be crushed by perspective foreshortening are omitted.

The series can also be viewed as travelling though the 120-cell, starting with a single dodecahedron at the “south pole” of the 120-cell. The trip continues through additional layers up to the “equator”. As we travel through the 120-cell, each shell we arrive at contains all the shells passed so far. After we cross the “equator” further shells resemble those we have travelled through already but in reverse order and contain all the shells we have passed through already. Eventually we will arrive at the “north pole”, a single dodecahedron that contains all the remaining 119 dodecahedrons of the 120-cell in its interior.

### Name conventions:

The two digit numbers in front of the name part "120-cell" is a consecutive number counting from lower to higher complexity. The L after “120-cell” stands for layer(s), followed by the numbers of the layers that contribute to the model. The number in front of the parenthesis denotes the layer number in units of dodecahedrons, the numbers between parenthesis in units of edges.A single dodecahedron has either 5, 6 or 7 layers of edges, depending on whether the vector of advancement crosses opposite faces, opposite corners or opposite edges.