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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

Schwarz P Egg

This egg design is part of a series of “10 Decorative Minimal Surface Eggs”. The series was originally created in 2021.
See my page for a more recent similar series of hollow “10 Easter Eggshells” that can be used to store objects or candy.
The descriptions below offer specific suggestions for printing. One general word of caution is when using “tree” supports, be sure to add “support blocks” to prevent the tree structures from extending up into the interior of the egg. They’ll be impossible to remove.
Expect to do a bit of post-print cleaning with each piece.
Schwarz' P Egg
This relatively simple egg requires a small amount of tree support to the bottom bulge. This is a good one to start with.
Schwarz’ D Egg
The Schwarz D Egg has some overhangs that may not be properly supported by standard supports. The piece is provided plain and with custom supports.
Schoen’s Gyroid Egg
The Schoen’s Gyroid Egg requires tree supports for the bottom third or so.
Schoen's Batwing Egg
The Schoen's Batwing Egg is tricky in that it has some rather large horizontal surfaces. The model shown was printed with an extensive amount of tree support. While the overall structure held up, the undersides of the horizontal surface required a lot of cleanup.
Fischer-Koch S Egg
The Fischer-Koch S Egg has some overhangs that may not be properly supported by standard supports. The piece is provided plain and with custom supports.
Schwarz’ CLP Egg
This relatively simple egg requires a small amount of tree support to the bottom bulge. This is a good one to start with.
Schoen’s I-WP Egg
Schoen’s I-WP Egg requires tree supports for the bottom third or so.
Schoen’s F-RD Egg
Schoen’s F-RD Egg has some overhangs that may not be properly supported by standard supports. The piece is provided plain and with custom supports.
Schoen’s F-RD(r) Egg
Schoen’s F-RD(r) Egg requires tree supports for the bottom third or so.
Fischer-Koch C(Y) Egg
Fischer-Koch C(Y) Egg has some overhangs that may not be properly supported by standard supports. The piece is provided plain and with custom supports.


Cube

Cube.stl

Clifford Sculptures (Carousel)

This sculpture is one in a series of three derived from the Clifford parallel geodesics of the cell-centered stereographic projection of the four-dimensional 600-cell. Visit my page for links to others in this series. See here for video: https://youtu.be/MTMeT03aBJ0 

The 600-cell is a stunningly beautiful regular 4-polytope, but its beauty is nearly impossible to perceive because, well, we can only perceive three dimensions!

One way to deconstruct and appreciate the 600-cell is to understand it in terms of geodesic great circles. The 600-cell is made up of 72 great decagons. The 72 great decagons can be divided into 6 sets of 12 non-intersecting Clifford parallel geodesics, such that only one decagonal great circle in each set passes through each vertex, and the 12 decagons in each set reach all 120 vertices.

When exploring or printing these sculptures, I encourage you to orient to the geometry by first locating the complete decagons. Every line portion of is course part of a larger great decagon that has been removed to reduce complexity and highlight specific aspects of the geometry.

These are advanced sculptures to print and require meticulous planning and support structures. The models pictured were printed with an FDM printer, 0.4mm nozzle at 0.1mm layer height, although these designs are probably better suited to SLA or SLS printing technology.

This project was inspired by the book "Visualizing Mathematics with 3D Printing" by Henry Segerman.

Learn more about Clifford parallel geodesics and 600-cell geometry here: https://en.wikipedia.org/wiki/600-cell 

Clifford Carousel

Named because of its circular symmetries, this sculpture is the easiest to print. It is made up of nine of the smallest Clifford parallel geodesics connected with “struts” to form three interconnected 30-tetrahedron Boerdijk–Coxeter helix rings. 

Clifford Core

Named because of its similarity to an apple core, this sculpture does not contain any full geodesic great circles, but instead shows the interrelation between portions of eleven geodesic great circles, one through the middle and ten twisting around. The sculpture can also be seen as portions of five twisted Boerdijk–Coxeter helix rings containing 21 of the full 30 tetrahedrons.

Clifford Climber

Named for is similarity to playground climbing structures, this sculpture contains nine of the largest Clifford parallel geodesics which, like the “Clifford Carousel,” are connected with “struts” to form three interconnected 30-tetrahedron Boerdijk–Coxeter helix rings. This sculpture is the most difficult to understand and by far the trickiest to print!


Ultimate Pasta

This is the ultimate piece of pasta. If you have a Pasta Printer you could print it for World Pasta Day on October 25 or US National Pasta Day on October 17.

See here for video: https://youtu.be/4qpxfTHBdzM 

This playful sculpture is derived from a matrix of Kelvin polyhedra, a fourteen-sided polyhedron with 6 squares and 8 hexagons. A triagonal piece of pasta is stuck on to each hexagonal face. This structure is also known as 8-srs. Yum.


Tetrahedron

Tetrahedron.stl

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.


Schoens F-RD(r) Egg

Lissajous Rope Knots

13,14,15,P,P,0.stl

Industrial Torus Stairs

Industrial Torus Stairs - With Supports.stl

Schoens Batwing Egg

Schoens I-WP Egg

Schoens F-RD Egg

Schoens F-RD Egg.stl

March Magnets - Day 7 #marchmagnets | Pie with pi symbol and crust with 3.14 (and more digits)

March Magnets_Day 7_Karen Chau Designs.3mf

Octahedron

Octahedron.stl

カタカナアルファベットサイコロ
Isohedral fair

Katakana Japanese Alphabet d48 Polyhedral Die

カタカナアルファベットサイコロ
Isohedral fair die with every Katakana symbol on its faces. Note that usually only 46 Katakana symbols are used. In this model two practically obsolete symbols ヰ and ヱ are used in order to obtain an isoheral 48-hedron

Katakana Japanese Alphabet 48-hedron

Schwarz CLP Egg

Reuleaux Tetrahedron

Reuleaux Tetrahedron.stl

math art

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Penta Tubes

120-Cell Sculpture: Six Intertwining Rings

Schwarz P Egg

Cube

Clifford Sculptures (Carousel)

Ultimate Pasta

Tetrahedron

120-Cell Rings

120-Cell Study

Schoens F-RD(r) Egg

Lissajous Rope Knots

Industrial Torus Stairs

Schoens Batwing Egg

Schoens I-WP Egg

Schoens F-RD Egg

March Magnets - Day 7 #marchmagnets | Pie with pi symbol and crust with 3.14 (and more digits)

Octahedron

Katakana Japanese Alphabet d48 Polyhedral Die

Schwarz CLP Egg

Reuleaux Tetrahedron