Symmetry & Combinatorics

Brief Notes on Symmetry & Combinatorics

by Frank C. Fung - 1st published in December, 2006.

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TABLE OF CONTENT
Topic #1Tri-Octo Symmetry
Topic #2Quad-Hexa Symmetry
Topic #3{ 24-Structural Symmetry }
Topic #4Multiplex Logical Symmetry
Topic #5Concluding Remarks

This paper is published in honor of my father, the late Charles Hok-Ling Fung ( 1909 - 2002 ) , on the occasion of his 97th birthday on 2006-12-26 .

Topic #1 --- Tri-Octo Symmetry :

Let us first bring-in a 3-D { Cube } , as per the diagram on-the-left , below .

Let us now construct a { 8-Pointed Star } , using the 8 corners of the cube as the { points } , as per the diagram on-the-right , above .

Let us now rotate the { 8-Pointed Star } about the { [ -1, -1, -1 ] - [ +1, +1, +1 ] } axis ,

The { 8-Pointed Star } is therefore also [ Tri-Symmetric ] about 4 axis :

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Topic #2 --- Quad-Hexa Symmetry :

Let us bring- in again a 3-D { Cube } , as per the diagram on-the-left , below .

Let us now construct a { 6-Pointed Star } , using the 3 orthogonal [ axis-directions ] , as per the diagram on-the-right , above .

Let us now rotate the { 6-Pointed Star } about [ X-Axis ] ,

The { 6-Pointed Star } is therefore also [ Quad-Symmetric ] about 3 axial-directions , i.e. :

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Topic #3 --- { 24-Structural Symmetry } :

Let us now bring-in a { X-Y-Z Reference Frame } , as per the diagram on-the-left , below :

Let as also identify six (6) fixed directions , as per diagram on-the-right , above , for mapping purposes .

There are then 24 ways that we can orient the { X-Y-Z Reference Frame } , mapping onto these 6 fixed-directions ,

Let us suppose that :

IF we found that the shape of the object remains constant throughout ,

THEN the object is said-to-be [ 24-Structural Symmetric ] .

And we offer three (3) such objects with this special property :

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Topic #4 --- Multiplex Logical Symmetry :

Let us now identify a set of eight (8) separate colors for our use in this section , namely :

Let us now bring-in a blank { 8-Pointed Star } to be colored , as per the diagram on-the-left , below .

Let us now paint each of the 8 arms of the { 8-Pointed Star } a different color , as per the 3 examples in the 3 diagrams on-the-right , above .

We simply note here that there are 40,320 ways , or [ 8-factorial ] ways , to do this ,

And these 40,320 configurations are said-to-be [ Logically Symmetric ] , because :

Let us now add a little twist & spice , by coloring the 8 { nodal points } the same color as the arm ,

Let us now rotate the entire [ object ] about the { "1"-axis } , as marked { grey-color } ,

Let us again rotate the entire [ object ] about the { "2"-axis } , as marked { grey-color } ,

Let us now try to arrive the configuration in { Diagram LS-3 } via the [ logical ] route , i.e. :

And the configurations in { Diagram LS-3 } & { Diagram LS-4 } are now the same . Interesting eh !

Let us try another configuration , as per the { Diagram LS-5 } on-the-left , below .

Let us now rotate the entire [ object ] about the { "1"-axis } , as marked { grey-color } ,

Let us again rotate the entire [ object ] about the { "2"-axis } , as marked { grey-color } ,

Let us now try to arrive the configuration in { Diagram LS-7 } via the [ logical ] route , i.e. :

And the configurations in { Diagram LS-7 } & { Diagram LS-8 } are now the same .

( Hint : try the [ 1-4-6-7 tetrahedron ] symmetry vs. the [ 8-5-3-2 tetrahedron ] symmetry . )

We also note here , in passing , that :

Symmetry , sometimes , can be more deceptive than first-meet-the-eye .

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Topic #5 --- Concluding Remarks :

The arithmetic equation for { Tri-Octo Symmetry } & { Quad-Hexa Symmetry } is quite naturally :

And we now make this observation :

The "I-CHING" may , therefore , have been rooted in the study of symmetry & assymetry of geometric shapes ,

And more effort in this direction may be warranted .

Finally , we bring-in the { Equilateral Tetrahedron } & the { 4-Pointed Star } :

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( Three Body Problem / Characteristics of Numbers / Matrix & Linear Algebra / 'I-CHING' / Triangle / Odd-On-Odd Format / Multiplication Table for Modulo P )

Original dated 2006-12-26 *** Updated 2008-11-01 / 2009-10-26