Jusur / Bridges Research Atlas

Let's Make a $(3^{6})_{\mathrm{D}}$ $(3^{6})_{\mathrm{L}}$ Chiral Tessellation Dance

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Let’s Make a Chiral Tessellation Dance

Year: 2012 Authors: Joseph D. Clinton

Core claim

A linkage of triangles can demonstrate how rotation and translation behave differently in theory and in a physically constrained model.

Topics

chiral tessellation, rotation and translation, hands-on workshop, physical modeling, animation

Domains

symmetry transformations, tessellations, geometric rotation, geometric translation, visualization, educational design, kinetic model, digital animation

Methods

model kit assembly, stepwise construction, POVRAY animation, frame-based rendering

Media

laser cut wood triangles, nylon push rivets, monofilament line, animated film

Paper text

The text below is the locally extracted OCR/Markdown version of the paper. Raw PDF files remain local and are not published here.

Bridges 2012: Mathematics, Music, Art, Architecture, Culture

Let’s Make a Chiral Tessellation Dance

Joseph D. Clinton Retired 334 Mackenzie Drive West Chester, PA, 19380-3816, USA E-mail: Jclinton@kennett.net

Abstract

Mathematics is an abstract language that often attempts to explain real world observations. Generalizations often emerge that become useful in explaining the physical world around us. Too often the abstract concepts are assumed to give precise solutions to physical realities. Hopefully, this workshop will give educators and their students’ hands-on experience to understand the differences between theoretical abstraction and the reality of applying physical restraints. Each participant in the workshop will receive a modeling kit to assemble and take with her/him. The model will illustrate the physical realities of applying the mathematical principles of rotation and translation transformations of a linkage of triangles from one symmetry form to another. Each participant will also receive an animated film that will illustrate the theoretical application of the same mathematical abstractions.

Introduction

Mathematics is one of the many areas of study that bridge all other areas of study and discovery. At one time it had become a universal language crossing the boundaries between the arts, sciences, humanities, technologies and other areas of specialization. However, in recent years, the study of mathematics has separated itself into many dialects becoming specialized languages of their own. Is the universal bridge of mathematics crumbling?

In the abstract world of mathematics, a point may be described as having zero dimensions. A line may be described as having one dimension only. A plane may be described as having two dimensions only. A solid may be described as having three dimensions. In the physical world, these abstractions cannot fully explain the behavior of the physical world. These definitions must account for the three dimensionality of physical reality to be able to fully describe its observed behavior.

The goal of the workshop is to give each participant hands-on experience to the differences between mathematical abstraction and the physical reality of applying rules of restraint to common materials.

The Workshop

Each participant in the workshop will receive a modeling kit to assemble and take with her/him. The model will illustrate the physical realities of applying the mathematical principles of rotation and translation transformations of a linkage of triangles from one symmetry form to another. Each participant will also receive an animated film that will illustrate the theoretical application of the same mathematical abstractions.

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Let’s Make a Chiral Tessellation Dance, the Physical Model

Required materials. The materials contained in the kit are: twelve - laser cut wood triangles, six - nylon push rivets, one - 8” length of diameter monofilament line, one - finger nail clipper and one - lighter.

img-0.jpeg Figure 1: Material required for model.

Step one. Choose two triangles. Push a rivet through one side of one of the triangles. Attach the other triangle with the rivet. Complete a total of six triangle assembly sets.

img-1.jpeg Figure 2: Step one - Two triangles & one rivet assembly.

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NOTE: The rivet should be placed on the same side of the assembled set for the desired symmetry of the final assembly.

Step Two. Choose two triangle assembly sets and a monofilament line. Push the line through one outside triangle and one inside triangle vertex. Heat the end of the line forming a ball larger than the hole - let cool. Pull the line tight and cut the other end of the line about inch long. Heat the end of the line forming a ball larger than the hole - let cool.

img-4.jpeg Figure 3: Step one - Six triangle assembly sets.

Let’s Make a (3^{6}){D} (3^{6}){L} Chiral Tessellation Dance

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img-8.jpeg Figure 4: Step two - Attach one inside triangle to one outside triangle of two sets.

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Step Three. Choose an adjacent vertex on the triangle assembly set and monofilament line. Push the line through one outside triangle and one inside triangle vertex. Repeat step two for all triangle sets and their matching holes.

NOTE: The choice of matching vertex holes will determine the direction of allowable rotation for all remaining sets.

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img-14.jpeg Figure 5: Step three - Attach an adjacent inside triangle to an adjacent outside triangle of two sets.

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Step Four. Enjoy the transformation of a chiral tessellation to a chiral tessellation.

img-17.jpeg Figure 6: Step four - chiral tessellation to chiral tessellation.

Let’s Make a Chiral Tessellation Dance, the Analytical Model

Description of Model. The analytical model is in the form of an animated film of a Chiral Tessellation. The center of rotation and translation for each set of triangles is located at the center of the six set tessellation. As with the physical model, the inside triangle of a set is rotated about its common outside triangle center. The two centers stay in common with each other throughout a full rotation of the respective triangle sharing the common center. The vertex of an inside triangle is connected to an adjacent outside triangle. A vertex connection is not allowed to be broken. The center of each set is translated along an axis through the center of the six set tessellation. In this model all triangles are equilateral.

Software Used. The animation and mathematical modeling used was version 3.6 POVRAY. [1] POVRAY is a freeware animation and graphics software package. The final animation was generated with version 5.2.2.173 AVS Video Editor. [2]

The POVRAY INI File. The POVRAY Chiral_Poly.ini file is an initialization file giving the command line code used to set the parameters for generating the animation frame files, in this case .jpg files.

; Chiral_Poly.ini Animation File Joseph D. Clinton 2012 version 3.62;

Antialias=On

Antialias_Threshold=0.1

Antialias_Depth=2

Input_File_Name=Chiral_Polygon.pov

Output_File_Name=FST36_.jpg

Initial_Frame=1

Final_Frame=360

Let’s Make a (3^6)_D (3^6)_L Chiral Tessellation Dance

// Subset_Start_Frame=1 // Subset_End_Frame=361 Initial_Clock=0 Final_Clock=1

Cyclic_Animation=on Pause_when_Done=off

The POVRAY POV File. The POVRAY Chiral_Polygon.pov file provides the code used to set the scene description for generating the animation.

// Chiral_Polygon.pov Joseph D. Clinton 2012 // POVRAY 3.62 Software #version 3.62; global_settings {assumed_gamma 2.1} #include “colors.inc” #include “shapes.inc” #include “textures.inc” #include “metals.inc” #include “skies.inc” #include “Transformation_36.inc” // include “TET_ELLIPSE_01.inc” light_source {< 0.0, 0.0, 0.0 > color rgb <1.000,1.000,1.000>} light_source {< 100.0, 100.0, 100.0 > color rgb <1.000,1.000,1.000>} light_source {< 00.0, 00.0, 100.0 > color rgb <1.000,1.000,1.000>} light_source {< -100.0, -100.0, -100.0 > color rgb <0.500,0.500,0.500>} camera {location (< 0.0, 0.0, 20.0 >) sky < 0.0, 0.0, 1.0 > look_at < 0.000,0.000,0.000 > angle 50} // background {rgb < 1.000, 1.000, 1.000 >} // White background {rgb < 0.000, 0.000, 0.000 >} // Black #declare CPTex1 = texture {pigment {color rgbt < 0.000, 0.000, 0.000, .3 >}} #declare Y1 = 0.0; #declare Y1 = 0.0; #declare Z1 = 0.0; #declare Axis = union { // sphere {< 0.0, 0.0, 0.0 > 0.5 texture {CPTex1}} // center of polygon set cylinder {< 5.0, 0.0, 0.0 >, < -0.0, 0.0, 0.0 > 0.02 texture {pigment {Red}}} cylinder {< 0.0, 5.0, 0.0 >, < 0.0, -0.0, 0.0 > 0.02 texture {pigment {Green}}} cylinder {< 0.0, 0.0, 4.0 >, < 0.0, 0.0, -0.0 > 0.02 texture {pigment {Blue}}} // object {Axis no_shadow rotate < X1, Y1, Z1 >} … object {TJ no_shadow translate < X1, Y1, Z1 > scale 0.13} // Chiral polygon

The POVRAY INC File. The POVRAY Transformation_36.inc file is an include file giving the mathematical operations and logic code used for generating objects to be animated.

// Transformation_36.inc Joseph D. Clinton 2012 // Rotation & Translation equations for chiral (3^6) // POVRAY 3.62 Software #version 3.62;

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// _______________ Rotation angle _______________

declare RATetB = clock*360; // Triangle rotation angle _ USE WITH Clock _______________

declare EndRATetB = RATetB+0.000001; // End Rotation cycles _ USE WITH clock _______________

// _______________ Rotation angle _______________

declare RATetA = 60/2; // One half triangle face angle

declare EL = 10.000000000; // Edge length of triangle

declare AL = 8.660254038; // Altitude of triangle (sqrt(EL^2-(EL/2)^2)

declare AL23 = 2*(8.660254038/3); // two thirds Altitude of triangle

declare RTet = 5.773502692; // (sqrt(3)/3)*10; // Center of triangle to vertex of triangle

declare XT0 = 0.0;

declare YT0 = 0.0;

declare YT1 = RTet;

declare AL23 = EL/2;

declare YT2 = RTet-AL;

declare XT3 = -EL/2;

declare YT3 = RTet-AL;

declare InT = 0.999;

declare OnT = 1.001;

// _______________

declare PETex1 = texture { pigment{color rgbt < 0.000, 0.000, 0.000, .3 > } }

declare PETex2 = texture { pigment{color rgbt < 0.000, 0.000, 0.000, .3 > } }

declare PETex3 = texture { pigment{color rgbt < 0.000, 0.000, 0.000, .3 > } }

declare PETex4 = texture { pigment{color rgbt < 0.000, 0.000, 0.000, .3 > } }

declare PETex11 = texture {pigment{color rgbt < 1.000, 0.000, 0.000, .8 > } } // { pigment{ Red } }

Bottom

declare PETex22 = texture {pigment{color rgbt < 1.000, 0.500, 0.200, .5 > } } // { pigment{ Orange } }

} Top

declare PErad1 = 0.15; // Radius of vertex

declare PErad2 = 0.1;

declare PEradc = 0.15; // Triangle center

declare Rrad = 0.05; // Radius of edge

declare RETex1 = texture { pigment{color rgbt < 1.000, 1.000, 1.000, .3 > } } // White

declare RETex2 = texture { pigment{color rgbt < 0.000, 0.000, 0.000, .3 > } } // Black

// _______________ Transformation of a 3.3.3.3.3.3 tessellation _______________

declare TJ = union {

//// _______________ Not used with Clock

////// declare RATetB = 0.000001; // start degree of rotation // not used with clock

// not used with clock

////// declare EndRATetB = 120.000001; // end degree of rotation // not used with clock

// not used with clock

////// while (RATetB < EndRATetB+0.000002) // operation // not used with clock

//// _______________ Not used with Clock

declare RATetC = 180 - (RATetA+RATetB); // Translation triangle angle C

declare CL = (RTetsin(RATetCpi/180))/sin(RATetA*pi/180); // Translation triangle length C

// declare CD = (RTetcos(RATetBpi/180)); // Center of triangle long leg along C

// declare TD = AL23-CD; // Center of triangle short leg along C

declare InT = 0.0001; // Top

declare OnT = -0.0001; // Bottom

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Let’s Make a (3^{6}){D} (3^{6}){L} Chiral Tessellation Dance

// Basic triangles (Top &amp; Bottom)
#declare X1 = -RTet*(sin(RATetB*pi/180));
#declare Y1 = RTet*(cos(RATetB*pi/180));
#declare Z1 = 0.0; // X1*tan(HDA*pi/180);
#declare TC1 = union { // Basic triangles (Top &amp; Bottom) at center
// sphere { <xt1, yt1,="" yt0,="" z1=""> PEradc texture { RETex1} rotate &lt; 0.0, 0, 0.0 &gt; } // Triangle Center
sphere { <xt1, yt1,="" z1=""> PErad1 texture { RETex1} rotate &lt; 0.0, 0, 0.0 &gt; } // Triangle vertex
sphere { <xt2, yt2,="" z1=""> PErad1 texture { RETex1} rotate &lt; 0.0, 0, 0.0 &gt; } // Triangle vertex
sphere { <xt3, yt3,="" z1=""> PErad1 texture { RETex1} rotate &lt; 0.0, 0, 0.0 &gt; } // Triangle vertex
cylinder { <xt1, yt1,="" z1="">, <xt2, yt2,="" z1=""> Rrad texture { RETex1} } // triangle edge
cylinder { <xt3, yt3,="" z1="">, <xt1, yt1,="" z1=""> Rrad texture { RETex1} } // triangle edge
cylinder { <xt3, yt3,="" z1="">, <xt1, yt1,="" z1="" z1=""> Xt2, YT2, Z1 &gt; &lt; XT3, YT3, Z1 &gt; } // Triangle face
} rotate &lt; 0.0, 0.0, 0.0 &gt; // Triangle face
}
// Basic triangles (Top &amp; Bottom)
#declare LD_Triangle = union {
object { TC1 translate &lt; 0.0, 0.0, OnT &gt; rotate &lt; 0.0, 0.0, 180+RATetB &gt; translate &lt; 0.0, CL, 0.0 &gt; texture { PETex11} } // Bottom
object { TC1 translate &lt; 0.0, 0.0, InT &gt; rotate &lt; 0.0, 0.0, 180-RATetB &gt; translate &lt; 0.0, CL, 0.0 &gt; texture { PETex22} } // Top
}
#declare FST_Triangle = union {
// First set triangles
object { LD_Triangle rotate &lt; 0.0, 0.0, 0.0 &gt; } // First set triangles
object { LD_Triangle rotate &lt; 0.0, 0.0, 60.0 &gt; } // First set triangles
object { LD_Triangle rotate &lt; 0.0, 0.0, 120.0 &gt; } // First set triangles
object { LD_Triangle rotate &lt; 0.0, 0.0, 180.0 &gt; } // First set triangles
object { LD_Triangle rotate &lt; 0.0, 0.0, 240.0 &gt; } // First set triangles
object { LD_Triangle rotate &lt; 0.0, 0.0, 300.0 &gt; } // First set triangles
}
#declare SST_Triangle = union {
object { FST_Triangle translate &lt; 0.0, (3*CL), 0.0 &gt; rotate &lt; 0.0, 0.0, 0.0 &gt; }
// Second set triangles
object { FST_Triangle translate &lt; 0.0, (3*CL), 0.0 &gt; rotate &lt; 0.0, 0.0, 60.0 &gt; }
// Second set triangles
object { FST_Triangle translate &lt; 0.0, (3*CL), 0.0 &gt; rotate &lt; 0.0, 0.0, 120.0 &gt; }
// Second set triangles
object { FST_Triangle translate &lt; 0.0, (3*CL), 0.0 &gt; rotate &lt; 0.0, 0.0, 180.0 &gt; }
// Second set triangles
object { FST_Triangle translate &lt; 0.0, (3*CL), 0.0 &gt; rotate &lt; 0.0, 0.0, 240.0 &gt; }
// Second set triangles
object { FST_Triangle translate &lt; 0.0, (3*CL), 0.0 &gt; rotate &lt; 0.0, 0.0, 300.0 &gt; }
// Second set triangles
}
object { FST_Triangle } // First set triangles
// object { SST_Triangle } // Second set triangles
// cylinder { <xt1, yt1,="" z1="">, <xt2, yt2,="" z1=""> Rrad rotate &lt; 0.0, 0.0, 180.0 &gt; translate &lt; 0.0, AL23, 0.0 &gt; texture { RETex2} } // triangle edges
// cylinder { <xt2, yt2,="" z1="">, <xt3, yt3,="" z1=""> Rrad rotate &lt; 0.0, 0.0, 180.0 &gt; translate &lt; 0.0, AL23, 0.0 &gt; texture { RETex2} } // triangle edges</xt3,></xt2,></xt2,></xt2,></xt1,></xt1,></xt1,></xt1,></xt1,></xt1,>
 
Clinton
 
// cylinder  $\{\langle \mathrm{XT3},\mathrm{YT3},\mathrm{Z1}\rangle ,\langle \mathrm{XT1},\mathrm{YT1},\mathrm{Z1}\rangle$  Rrad rotate  $&lt;  0.0$  0.0,  $180.0&gt;$  translate  $&lt;  0.0$  AL23,  $0.0&gt;$  texture  $\{\mathrm{RETex2}\} \}$  // triangle edges
// Not used with Clock
//// #declare RATetB = RATetB + 5; // increment degree of rotation // not used with clock
//// #end // end of loop // not used with clock
// Not used with Clock
}
// End of Transformation of
 
Animation Frames. The animation rotates the vertices of the triangle sets through a complete  $360^{\mathrm{o}}$  cycle.
 
![img-18.jpeg](images/img018.jpeg)
Figure 7: Animation frames  $(3^{6})_{\mathrm{D}}(3^{6})_{\mathrm{L}}$  chiral tessellation,  $0^0$  to  $360^0$  at  $15^0$  intervals.
 
Discussion. An open discussion will be held after viewing the analytical model and constructing the physical model. The discussion will focus on limitations between the two models.
 
[1] POVRAY, POV-Team, 2002. Persistence of Vision (TM) Ray Tracer. POV-Team, Williamston, Australia. http://www.povray.org/
[2] AVS Video Editor, Online Media Technologies Ltd. 2011. http://www.AVS4You.com

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