Perimeter Magic Platonic solids

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Index

Introduction Prefix The Cube
The Tetrahedron The Octahedron The Icosahedron
The dodecahedron The Icosadodecahedron The Buckyball
  References  

Introduction

Bao Qi-shou lived in the latter part of the nineteenth century. He was an unsuccessful businessman, a soldier, a satirical poem writer, and finally a student of astronomy and mathematics and a teacher of mathematics to gifted students. He self-published a book in about 1880 called Bi-Nai Mountain Hut Records which included an article Zen Bu Suan Fa Hun Yuan Tu (Augmented Solid and Spherical Mathematical Diagrams) [a][1].

It is that article (which appeared as volume 2 of Bao’s book) that is the subject of this web page. The 24 pages of that volume are an appendix to Lih (1986 [2]) and are shown on this page.

I first became aware of Bao’s work when I received an email from Tung Hsu in August of 2010. He mentioned that his mothers Grandfather (Bao Qi-Shou) had found magic perimeter solutions for the five platonic solids in the late 1800’s. He later sent me some supporting documents [2], [4], and several photographs.
I contacted Dr. Ko-Wei Lih and received helpful suggestions and material from him.

[4]

My Sources:
  1. Lih, Ko-Wei, Bao Qi-shou and his Polyhedral Hun Yaun Yu. 1993, Cheng-hung Lin and Daiwie Fu (Eds.), Philosophy and Conceptual History of Science in Taiwan, pps 209-220.
  2. Lih, Ko-Wei, 論保其壽的渾圓圖, 1986, 22 pages, All text in this paper is Chinese, but it contains images of all pages from Vol. 2 of Bao.

I am indebted to Dr. Ko-Wei Lih for providing me with both of these papers. Without them this web page could not have been written.

  1. Contains short English language descriptions of most of the pages in Bao’s volume 2 along with translations of many diagrams.
  2. Contains images of all 24 pages of Bao’s volume 2.

More elaborate descriptions of these two papers, as well as other references, are located at the end of this page.

And , of course, I greatly appreciate Tung Hsu bringing this subject to my attention. Tung is interested in solving the problem of assigning numbers 1 to 60 to the vertices of the buckyball, so faces are magic. Is this possible? Suggestions anyone?

Above; Bao's two volumes

Below; Cover of volume 2

[1]

My text relies heavily on the brief notes in Lih (1993) [1], as I found that the Web translation services did not produce useful results. I found interesting the fact that Bao used different methods (3-dimensional line drawings, planar graphs, and diagrammatic illustrations) to illustrate his figures!

These are the only pages in Bao's volume 2 that do not contain illustrations.
They are pages 1, 2, and 24.

I include them for the benefit of any possible Chinese language readers.

 

 

[a] Lih (1993, [1]) quoting an earlier research paper of 1935.

Prefix

Types of perimeter magic:

  • Vertex V-magic The numbers at the vertices of each surface (face) sum to a constant. If V-magic, then F-magic (if no numbers are on the sides).

  • Edge E-magic the numbers along each edge of the graph sum to a constant. If E-magic then F-magic, but F-magic does not require E-magic. i.e. all edges may not sum the same

  • Face F-magic the sum of all the lines around the perimeter of the face equals a constant.

These designations were first proposed by Charles Trigg in numerous articles on the subject written in the 1970’s. [5][6] Although Lih uses a different terminology in his papers, I have chosen to use Trigg’s, so as to be consistent with my other pages.

The Platonic Solids:

Cube
V=8, E=12, F=6

Tetrahedron
V=4, E=6, F=4

Octahedron
V=6, E=12, F=8

Icosahedron
V=12, E=30, F=20

Dodecahedron
V=20, E=30, F=12

Two other shapes (not platonic) will be illustrated and discussed later.
The icosadodecahedron consists of 12 pentagonal faces and 20 triangular faces (the six-banded sphere).
The Buckyball is a round structure consisting of 12 pentagons and 20 hexagons with a total of 60 vertices (carbon atoms).

The Cube


Page 4


Page 5


Page 6


Page 7

Page 4 --This 3 dimensional figure shows numbers 1 to 8 arranged in a vertex face magic configuration such that each face sums to 18.

Page 5 --This figure shows the edges labelled to form a face magic cube. Numbers used are 1 to 12. Each face sums to 26.

Page 6 -- This fully perimeter magic cube is obtained by combining the previous edge and vertex magic figures. In this case the range of numbers from 1 to 12 have been increased by the constant 8 to provide a range of unique numbers from 1 to 20. Each face sums to 76.
The lower range of numbers could have been increased in value instead (by the constant 12). The face constant then would have been a larger number.
Note that in Boa’s original diagram, the label 2 on the top back edge should have been a 20.

Page 7 --This drawing illustrates the principle that complementary pairs may be inserted on the edges of a vertex face magic cube to produce an order-4 perimeter magic cube. Here the vertices are the series from 1 to 8. The edges use the numbers from 9 to 32 in complement pairs each summing to 41. Each face sums to the constant 182.
See my previously posted reference to this figure here. There C. Pickover improved on Bao’s figure by rearranging the interior edge numbers so each edge sums to 50 (with the face constant still 182).

The Tetrahedron


Page 8


Page 9


Page 10


Page 11

Page 8 --Uses the numbers 1 to 12 in complement pairs to label the edges of the tetrahedron, so the faces all sum to 39.

Page 9 --This planar net figure uses numbers 1 to 4 placed on the vertices and the faces labelled with numbers 5 to 8. The result is face magic with S=14. Strictly speaking, this is not perimeter magic.

Page 10 --This 3-dimensional drawing of a tetrahedron shows the same arrangement of the numbers 1 to 8.

Page 11 --This planar net figure uses numbers 1 to 16 and is face magic with S=72. My translation is a 3-dimensional drawing.

Page 12 --shows the figures of pages 9 and 11 but labelled with the complements of those numbers. i.e. 8 replaces 1, 7 replaces 2, etc. in the first figure; 16 replaces 1, 15 replaces 2, etc. in the second figure.


Page 12
In the text on these pages Bao explains how, by trial and error, it is easy to see that there is no way to produce constant face sums with consecutively labelled edges. [1, p, 214]

 

 

To the right is the translation of Bao's page 12 images.

  

The Octahedron


Page 3

 

 

Page 3 --Two representations based on the same consecutive vertex labelling. [1, pp 214-5]

I decided not to attempt a translation of these figures.


Page 13

Page 14

Page 13 --Bao specifically discusses the method of switching to complements. i.e. each number switched to the difference between it and 19. This results in all faces summing to 66. Lih [1 p.314] comments that Bao also mentioned substituting numbers 1 to 12 with 7 to 18, and 13 to 18 changed to 1 to 6. But this didn't wok for me! It results in an octahedron with 5 faces summing to 48, two summing to 66, and one equal to 84!
As shown, this is an order-3 perimeter E-magic octahedron using numbers 1 to 18 with S = 48.

Page 14 --Edge labelling of complement pairs. This figure uses complement pairs of the numbers 19 to 42. So each edge sums to 61 making the figure edge magic with each face sum 183.As each pair sums to the constant 61, the pairs may be placed on any edge of the previous figure to produce an order-5 perimeter magic octahedron.

The Icosahedron

Page 15 --The consecutive numbers 1 to 72 are placed so that the faces of all 20 triangles sum to 279. Bao’s page 15 has 2 numbers missing. 22 and 50 should appear between 9 and 15. This is corrected in Lih’s translation drawing.


Lih's translation of pae 15 [2, fig. 5, p.7]

Page 16 --Lih [1, p 215] mentions only that Bao has one mistake on this page. The number 24 between 70 and 2 should be 80.

He mentions nothing else about what this diagram represented.

The dodecahedron


Page 17

Page 19
I find it interesting that Bao used a variety of illustrations to demonstrate his perimeter magic figures.
For the dodecahedron he uses a planar net drawing on page 17, then on pages 18 and 19 he uses 3 dimensional illustrations rather then line drawings as on some previous pages.

Page 17 -- A planar graph representation with vertices consecutively labelled from 1 to 20, Faces are anti-magic. The 5 vertex numbers around each pentagon face sum to consecutive numbers from 47 to 58. So this figure is V-antimagic.

Page 18 -- Presumably Bao used this illustration to emphasize that it is impossible to have constant face sums using consecutive vertex labelling.

Page 19 -- My version of Bao's 3-dimensional illustration shows the numbers on the back side upside down, as does Bao's.
This figure is Face magic: S = 230. It uses consecutive numbers from 1 to 50.
The vertices use consecutive numbers from 1 to 20 (same as bao-17).
When the vertex numbers (1 to 20) are removed and the edge numbers reduced by 20, the figure is E-antimagic, with sums equal to the consecutive numbers from 72 to 83.

 
Page 18

The Icosadodecahedron


Page 20

Page 21

Page 22

Page 23

Page 20 -- Lih [1] p.218 says there is no indication of how Bao arrived at this complicated configuration. Numbers 1 through 90 are assigned so that six numbers around each of the 20 triangles add up to 228 and the 10 numbers around each of the 12 pentagons add up to 380. The vertices are labelled with the consecutive numbers 1 to 30. (I show a translation below.)

Page 21 -- This figure is obtained by adding 60 to each vertex and subtracting 30 from each edge of the page 20 figure. The result is then reflected vertically. [1] p.218

Page 22 -- This is the complementary numbering of page 20. [1] p.218

Page 23 -- This image is of a ball wrapped with 6 intertwined paper bands. If these are regarded as edges, we have a combined Icosahedron and Dodecahedron. Thus it has 20 triangle faces and 12 pentagon faces. Below is shown Tung Hsu’s model of this figure with the numbers from the planar diagram of page 20.

Page 24 -- (Shown at the top of this web page) is translated by Lih [1] as follows:

Further diagrams can be constructed by the previous methods. It becomes easier to make variations when more and more numbers are used. However, it is pointless to give these illustrations when all methods have been exhaustively demonstrated.


Page 20

[4]

The Buckyball

The Buckyball is a round structure consisting of 12 pentagons and 20 hexagons with a total of 60 vertices (carbon atoms). The C60 carbon molecule, which is harder then diamond, has a carbon atom at each vertex of this structure.

If this structure looks familiar, it probably is. The most popular design of soccer ball is the 32-panel.
It uses this same 12 pentagon and 20 hexagon pattern.


[4]

Tung Hsu, Bao's Great Granson, is interested in the problem of assigning a number at each vertex of this structure.

Is it possible to assign the consecutive numbers from 1 to 60 in such a manner that all 12 pentagon faces sum to a constant, and all 20 hexagon faces sum to a constant?

The figure on the right is a 2-D version. If working on this problem, it may be helpful to print out an enlarged version of this image to help in assigning numbers.

References

  1. Lih, Ko-Wei, Bao Qi-shou and his Polyhedral Hun Yaun Yu. 1993, Cheng-hung Lin and Daiwie Fu (Eds.), Philosophy and Conceptual History of Science in Taiwan, pps 209-220. 12 page English paper on Bao
    It contains a short biography of Bao Qi-Shou and a very brief history of magic squares in China. The author then presents a brief description of most of the pages of Bao’s book along with some translated diagrams . He also refers to an earlier translation of Bao’s work by Li Yan in his 1935 article A Study of Chinese Mathematicians’ Magic Squares.
    It is available for downloading from this site as Bao Qi-shou.pdf.

  2. Lih, Ko-Wei, 論保其壽的渾圓圖, 1986, 22 pages, All text in this paper is Chinese, but it contains images of all pages from Vol. 2 of Bao. It is available at http://episte.math.ntu.edu.tw/articles/ar/ar_li031207_1/page2.html . I have used these images (with permission) on my page.

  3. Lih, Ko-Wei, On Magic and Consecutive Labelling of plane graphs, 1980, 33 pages, , Utilitas Mathematica, 24(1983), p.165-p.197. No mention of Bao, but lots of magic labelled graphs.
    I have not used material from this paper, but include the reference for it's relevance.
    It is available for downloading from this site as Graph Labelling.pdf.

  4. Tung Hsu, "An introduction to Bao Qi-shou and his 'Suan Fa Hun Yuan'" (in Chinese), Bulletin of the Association for the History of Science, No. 13, 2009, pp. 85 - 90. It contains an image of the two volumes of Bao’s essay.

  5. Charles W. Trigg, Second Order Perimeter-magic and Perimeter Anti-magic Cubes, Mathematics Magazine, 47(3), 1974, pp 95-97.

  6. See more information and other references on my other 3 perimeter.htm pages.

Many thanks again to Professor Tung Hsu and Dr. Ko-Wei Lih!
Tung Hsu is a recently retired physics professor now living in the United States.
Dr. Ko-Wei Lih, is a mathematician with Institute of Mathematics, Academia Sinica, Taiwan.

This page was originally posted Sept. 27, 2010
It was last updated November 02, 2010
Harvey Heinz   harveyheinz@shaw.ca
Copyright © 1998-2009 by Harvey D. Heinz