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

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.

Beadworks for the mathart exhibition (Mar. 15, 2014)

From pentaexcavated Icosidodecahedron to Mackay polyhedra

Quasicrystal (準晶)

The introduction to quasicrystal in Zometool website

I saw a page at zomeool company website about the quasicrystal model made by the zometool. There is a paragraph about the model displayed in the 2nd floor, outside the Pan-Guan Lecture Room of the chemistry departmen, National Taiwan University. My former student, Hsin-Yu Ko, kindly introduced the model to the students from the King David School in Taipei, possibly early last summer.

http://zometool.com/goto/quasicrystals

The history of the zometool model of this quasicrystal is here.

準晶模型的前世與今生 The evolution of the 3D quasicrystal model

I wrote something in my facebook about the origin of the zometool model of quasicrystal (mainly in Chinese with a little bit English) which my students and I designed and created :

準晶模型的前世與今生 The evolution of the 3D quasicrystal model

Helen Yu, the dealer of zometool in Taiwan, also posted a lot of photos about the reconstruction of this model together with the carbon onion at the National Taiwan Science Education Center here:
Gluing& Hanging Models (Photo Credits)- keeping uploading..

Zometool: carbon onion and quasicrystal almost done

Photos by Helen Yu.

Zometool: quasicrystal and carbon onion

National Taiwan Science Education Center (NTSEC, 國立臺灣科學教育館) is going to have new area for the chemistry demonstration. They also decided to hang the two zometool models, carbon onion and quasicrystal, we designed, constructed and donated (Chemistry department, National Taiwan University). However, the original structures, particularly the quasicrystal, are not stable enough to be hung under the ceiling. So we decided to strengthen the original quasicrystal model by adding a stick along the short diagonal of each face. These sticks are blue, which, I believe, can make the the whole model much more colorful. Additionally, my students have also carefully glued each part together. It takes a lot of work. So, the whole project is still unfinished yet.

From left to right: 郭岷翔，范原嘉（Yuan-Jia, Fan），詹欣穆，秦逸群，黃泓穎 (photo by Helen Yu)

Quasicrystal zometool model donated to the National Taiwan Science Education Center (NTSEC, 國立臺灣科學教育館).

A few months ago, I decided to donate the two zometool models of quasicrystal and six-layer carbon onion to the National Taiwan Science Education Center (NTSEC, 國立臺灣科學教育館). Finally, Hsin-Yu, Yuan-Chia and some other students of mine moved these two models to the NESEC last week.

Originally, due the the tight budget, we couldn't really make the complete zometool model of quasicrystal consisting of stellated rhombic triacontahedra up to the second level. Fortunately, Ms. Helen Yu, the owner of a local Kindergarden, kindly supported us to finish this two-level quasicrystal structure. Here are a few photos from the special activity with kids from the Helen's Kindergarden.

Yuan-Chia and Hsin-Yu explained the construction rules to Kids:

Kids were working on the basic units:

Hsin-Yu, Bang-Rui, Yuan-Chia, I-Chun, and Hsin-Mu took a picture together in front of this beautiful zometool model of quasicrystal:

An old photo from Paul Hildebrandt

I got a nice picture of D. Shechtman with Zometoy from Paul Hildebrandt, President of Zometool Inc:

(NIST, 1985)

Zometool models at the Quasicrystal international conference

The newest Nobel laureate in Chemistry, D. Shechtman, was invited to Taiwan to participate a quasicrystal international conference held in the National Taipei University of Technology (May 7-9), a university very close to the National Taiwan University where I am working.
Yuan-Chia, Qian-Rui, and Hsin-Yu moved the two large zometool models, a 3D quasicrystal consisting of two types of golden rhombohedra (prolate and oblate) and a six-layer carbon onion (C20, C80, C180, C320, C500, and C720), to the conference's lobby area. Most participants seem to enjoy these two types of quasicrystal models, cluster and tiling, a lot.

Prof. Shechtman took a picture of me in front of the zometool model of carbon onion.

I also gave Prof. Shechtman a bead model of high-genus fullerene, a topologically nontrivial quasicrystal :-), as a gift.

Quasicrystals with Zometool

Yuan-Chia Fan (范原嘉), Hsin-Yu Ko(柯星宇) and Mu-Chieh Chang (張慕傑) made a beautiful quasicrystal consisting of two types of rhombohedra with zometool last Friday. They put this model in the main lobby of our chemistry department for the alumni reunion last Saturday.

In addition to the tiling approach to the quasicrystal, one also constructs it using the cluster approach. In this sense, the carbon onion we made before is also a quasicrystal.

Originally, we didn't order enough red sticks to connect different shells of this structure. So only a few red sticks are used to hold neighbored Goldberg polyhedra along one five-fold axis (z direction). We now have enough red sticks for all twelve (or six) 5-fold axes emanating from the central dodecahedron (Goldberg vector (1,0)). So libration motion of each shell is quenched.

Yuan-Chia, Hsin-Yu, and Mu-Chieh also designed a wonderful support around pentagon at the bottom to enhance the stability for the this six-layer Goldberg polyhedra.