Da Vinci's elevated polyhedra

Leonardo da Vinci (1452-1519) made outstanding illustrations for Luca Pacioli's 1509 book "The Divine Proportion", in which they described "elevated" forms of many polyhedra. In the Seoul Bridges meeting this year, Rinus Roelofs presented a beautiful paper on the similarities and differences between Da Vinci's elevations and Kepler's stellations.

For details, check the following pdf file:

Rinus Roelofs, Elevations and Stellations, Proceedings of Bridges 2014: Mathematics, Music, Art, Architecture, Culture, 235-242.

Figure 1 and 2 in the paper are original illustrations made by Leonardo da Vinci:

It is interesting that bead models for the five elevated regular polyhedra can be built easily with great effects. Among them, elevated cube and dodecahedron are more flexible as expected.

Also, these five elevated Platonic solids can be viewed as nonconvex deltahedra with the names, triakis tetrahedron, tetrakis hexahedron, triakis octahedron (stella octangula), pentakis dodecahedron, and triakis icosahedron, respectively.

Additionally, the elevated icosidodecahedron was also illustrated beautifully by Da Vinci in the book.

The corresponding bead model can also be built!

Frequency 2 Icosahedron

One more Truss model of perovskite (or fluorite/antifluorite) structure

I made another truss model of perovskite structure with octahedra constructed by yellow and blue bugle beads.

Octahedral diamond net

Truss model of perovskite (or fluorite/antifluorite) structure

Spinel structure

Truss models of cubic and hexagonal closest packings

I made two more bead models of two-layer cuboctahedron and twisted cuboctahedron (anticuboctahedron) to illustrate the vector equilibrium of Buckminster Fuller. Through these two models, the connection to the cubic and hexagonal closest packings can also be visualized more easily.

Two icosahedral complexes derived from an icosahedron

Starting from a bead model of icosahedron, one can make a few beautiful rigid polyhedral complexes by adding more regular octahedra and tetrahedra surrounding the central icosahedron. Here are two examples:

Icosahedron + Icosidodecahedron

Icosahedron + Icosidodecahedron + Rhombic Hexecontahedron

Tetrahelices - two bead models of Boerdijk–Coxeter helix

There is a helical tower based on the Boerdijk–Coxeter helix in Mito-shi, Ibaraki-ken, Japan:

Art Tower MIT

So, during my visit to Nagoya this May, I made several tetrahelices as souvenirs for my friends there. Since I brought many tubular beads with different colors to choose from, so I decided to use different color for three helical edges in each tetrahelix. In total, there should be four possible arrangements or two enantiomer pairs. But I only made two of them shown in the following picture. After I made these two tetrahelices, so many people also like to have this model, so I made two more tetrahelices before I left Japan.

Sierpinski tetrahedron & Mackay icosahedron for the Bridges Seoul 2014

Dr. Tsoo and I have also submitted two beadworks for the mathart exhibition in the Bridges Seoul 2014, which can be found here: tubular bead structures.

Polyoxometallate - Dowson structure

Octahedra connected by octahedral tunnels (38)

Bead model of Kaleidocycle (萬花環)

Kaleidocycle or a ring of rotating tetrahedra was invented by originally by R. M. Stalker 1933. The simplest kaleidocycle is a ring of an even number of tetrahedra. The interesting thing about the Kaleidocycle is that you can twist it inwards or outwards continually. The geometry of kaleidocycle has been studied by many people from different fields in the last 80 years:

1. Stalker, R. M. 1933 Advertising medium or toy. US Patent 1,997,022, filed 27 April 1933 and issued 9 April 1935.
2. Ball, W. W. Rouse 1939 Mathematical recreations and essays, 11th edn. London: Macmillan. Revised and extended by Coxeter, H. S. M.
3. Cundy, H. M.; Rollett, A. R. 1981 Mathematical models, 3rd edn. Diss: Tarquin Publications.
4. Fowler, P. W.; Guest, S. Proc. R. Soc. A 461(2058), 1829-1846, 2005.
5. 全仁重, Motivation Behind the Construction of Maximal Twistable Tetrahedral Torus.
6. HORFIBE Kazunori, Kaleidocycle animation.

Typically, people use paper or other solid materials to make this kind of toy. A few months ago, I discovered that you can easily make this toy by tubular beads through the standard figure eight stitch (right angle weave).

This particular model consists of 8 regular tetrahedra. You can easily extend rings that contain 10, 12, ... tetrahedra.

The procedure I used to make this 8-tetrahedra Kaleidocycle is by the standard figure-eight stitch (right angle weave) in which one just keep making triangles. Of course, some care should be paid on the sequence of these triangle.

ε Keggin structure

δ Keggin structure

γ Keggin structure

β Keggin structure

α Keggin structure

From pentaexcavated Icosidodecahedron to Mackay polyhedra

Perovskite, 4 stella octangula