Chain Link Klein Bottle

This model uses chain link latticing to represent a three-dimensional approximation of the four-dimensional “Klein Bottle.”

See here for video: https://youtu.be/AWJ-R3GBT80 

The concept of a Klein bottle was first described in 1882 by the German mathematician Felix Klein. It is a closed non-orientable surface that has no inside or outside. It is similar to the Möbius strip, but unlike the Möbius strip it has no boundaries and cannot physically exist in three-dimensional space.

In this three-dimensional representation, the surface intersects with itself, but in four-dimensional space the self-intersection can be eliminated. Unfortunately, I don’t have a 4D printer yet.

Modelling this intriguing shape in chain link latticing allows for visualization of the whole surface.


120-Cell Polychora Planter

This Polychora Planter is derived from the four-dimensional 120-cell 4-polytope, or polychoron.

The 120-cell is one of six regular convex four-dimensional shapes. It is composed of 120 dodecahedral cells. 

4-polytopes cannot be seen in three-dimensional space due to their extra dimension, but there are several techniques that can be used to “project” 4-polytopes into three-dimensional space. This design uses the cell-centered stereographic projection method, chosen for the interesting aesthetic appearance.

While these three-dimensional shapes have curved faces and edges of various lengths, it is interesting to imagine that in four-dimensional space, the faces are flat, and the edges are straight and identical in length.

In this design, select cells have then been removed to allow better visualization of the inner cells, and the outer cells have been hollowed for uses as planters of containers of some kind.


120 Cell.stl

This is a three-dimensional shadow of a four-dimensional shape called the “120-cell.”

See here for video: https://youtu.be/oquFgOw6k28 

The 120-cell is one of six regular convex four-dimensional shapes. It is composed of 120 dodecahedral cells.

Technically, this model only includes 107 of the 120 dodecahedral cells. The outer layer of cells is proportionally much larger and would be difficult to print. In fact, the outmost cell is infinitely large, which would be very difficult to print.

This is a cell-centered stereographic projection, meaning that there is a single small dodecahedral cell at the center. Can you see it?

This model is designed for resin printing.


120-Cell Sculpture: Two Intertwining Rings 

120-cell Two Intertwined Edge Rings.stl

120-Cell Sculpture: Two Solid Rings 

120 Cell Sculpture Two Solid Rings.stl

120-Cell Sculpture: Six Intertwining Rings

This sculpture highlights the edges of sets of intertwining rings related to Hopf fibrations within the four-dimensional 120-cell 4-polytope.

Because the shapes exist in four-dimensional space, special mathematical techniques are required to visualize them. These sculptures are made using the cell-centered stereographic projection method to create three-dimensional “shadows” of the original four-dimensional shapes. 

The 120-cell is one of six regular convex four-dimensional shapes. It is composed of 120 dodecahedral cells and can be partitioned into twelve 10-cell great circle rings related to the Hopf fibrations.  The lines in this sculpture outline the edges of six of the intertwining rings, for a total of 60 cells.


120-Cell Edge Sculpture

120-Cell Rings

120-cell Linked Rings.stl

120-Cell Study

The 120-cell is a four-dimensional shape composed of 120 dodecahedral cells. It is one of the six regular convex 4-polytopes and has some of the most beautiful patterns and symmetries.

4-polytopes cannot be seen in three-dimensional space due to their extra dimension, but there are several techniques that can be used to “project” 4-polytopes into three-dimensional space. This study uses the cell-centered stereographic projection method, chosen for the interesting aesthetic appearance.

While these three-dimensional shapes have curved faces and edges of various lengths, it is interesting to imagine that in four-dimensional space, the faces are flat, and the edges are straight and identical in length.

In this 120-cell study, select cells have been removed to allow maximal viewing of the various cells, including the center cell. The outer “shell” of cells has been removed to allow better visualization of the inner cells, and because their relative size is so large compared to the inner cells.


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Chain Link Klein Bottle

120-Cell Polychora Planter

120 Cell.stl

120-Cell Sculpture: Two Intertwining Rings

120-Cell Sculpture: Two Solid Rings

120-Cell Sculpture: Six Intertwining Rings

120-Cell Rings

120-Cell Study