Poly-Twistor by 3D Printer Classification of 3D Tori
Year: 2013 Authors: Akio Hizume; Yoshikazu Yamagishi; Shoji Yotsutani
Core claim
Poly-Twistors form a parametric family of 3D torus-like structures that can be generated as STL data and fabricated as physical models by 3D printer.
Topics
3D printed topology, torus knots, polyhedral symmetry, chirality
Domains
topology, knot theory, geometric symmetry, parameterized surfaces, 3D printing, computational design, sculptural form, rapid prototyping
Methods
geometric classification, integer frequency parameterization, C program generation, STL fabrication
Media
plastic, STL data, CG renderings, 3D printer
Paper text
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Proceedings of Bridges 2013: Mathematics, Music, Art, Architecture, Culture
Poly-Twistor by 3D Printer Classification of 3D Tori
Akio Hizume, Yoshikazu Yamagishi, Shoji Yotsutani Department of Applied Mathematics and Informatics Ryukoku University Seta, Otsu, Shiga, Japan E-mail: akio@starcage.org
Abstract
Using 3D printing, we can manufacture forms called Poly-Twistors. These are knotted closed loops around the surface of either a torus or a polyhedron. Variants have handedness and can be parameterized using different integer winding frequencies.
cube
octa
dodeca
icosa
rhomb30
Tri
Tetra
Hexa
Deca
XV
Figure 1: The Five Simplest Poly-Twistors
Figure 2: Real Models of the Simplest Poly-Twistor Size: from to diameter, Material: plastic
We can consider the three-dimensional torus by identifying the parallel aspect to face each other of the polyhedron. The Poly-Twistor is one expression of the 3D tori. We show the five simplest Poly-Twistors and their corresponding polyhedron on Fig. 1. We can produce the real models by 3D printer as shown on Fig. 2.
Hizume, Yamagishi and Yotsutani
Figure 3: An Example of Helical-Torus in case of
Figure 4: plus-minus chirality
The Poly-Twistor is an assemblage of identical helical-torus arranged with polyhedral symmetry. We define any helical-torus by a single frequency. We show an example in Fig. 3. When the frequency is a fraction (not integer), the helical-torus makes a torus knot. We can decide any frequency, amplitude of helices, thickness of the tube, and cross-section.
There are two kinds of chirality governing any two helical-tori interlock with each other. We define left type is “+ (plus)”, and right type is “- (minus)” as shown on Fig. 4. Tri-Twistor and XV-Twistor don’t have such
chirality because tori’s equatorial planes intersect in a right angle when we view from the 2-fold rotational axis of symmetry. Therefore there are eight kinds of Poly-Twistor. Further technically, there are sixteen kinds of Poly-Twistors because helical-torus itself has also chirality of clockwise and counterclockwise.
Tetra-Twistor 3/1 minus
The simplest Tetra, Hexa, and Deca-Twistor in Fig. 1 are plus chirality. We show minus simplest in Fig. 5.
We wrote the C program to generate STL data of the Poly-Twistors directly. We can produce almost endless kinds of such topological structures. We show some sample CG and photos of real models from next pages.
Hexa-Twistor 5/1 minus
Deca-Twistor 3/1 minus
Figure 5: Three minus simplest of Poly-Twistores
Poly-Twistor by 3D printer: Classification of 3D Tori
Tri-Twistor 2/1
Tetra-Twistor 3/2 plus
Tetra-Twistor 3/1 minus
Hexa-Twistor 5/2 plus
Hexa-Twistor 5/1 minus (trianglar section)
Figure 6 (1): Samples of Poly-Twistors.
Hizume, Yamagishi and Yotsutani
Hexa-Twistor 5/2 plus
Deca-Twistor 3/2 minus
XV-Twistor 2/1
Figure 6 (2): Samples of Poly-Twistors.
References
[1] Akio Hizume, Hexa-Twistor, MANIFOLD #01, pp. 10-12, 2000. (in Japanese) [2] Akio Hizume, Poly-Twistor, MANIFOLD #04, pp. 8-9, 2002. (in Japanese) [3] Akio Hizume, Poly-Twistor, ISAMA Proceedings, 2002. (This paper was reprinted on “inter-native architecture OF music”, Star Cage Publishing, pp. 199-204, 2006.) [4] Akio Hizume, Hexa-twistor Trianglar Section, MANIFOLD #05, 2002. [5] Akio Hizume, Real Model of the POLY-TWISTOR Triangular-Section, MANIFOLD #06, 2003. [6] Akio Hizume, Hexa-twistor Trianglar Section, 6th Bridges Proceedings, 2003. [7] Akio Hizume, Yoshikazu Yamagishi and Shoji Yotsutani, Poly-Twistor for Rapidprototyping, MANIFOLD #22, pp. 5-10, 2012. (in Japanese)