The Obtetrahedrille as a Modular Building Block for 3D Mathematical Art

Year: 2019 Authors: Tom Verhoeff; Koos Verhoeff

Core claim

The obtetrahedrille is a versatile space-filling module for constructing 3D mathematical art, including beam systems and sculptural forms.

Topics

modular geometry, 3D mathematical art, lattice structures, turtle notation, magnetic assembly

Domains

polyhedra, lattice geometry, tessellations, constant-torsion polygons, sculpture, generative design, mathematical art, metal fabrication

Methods

3D turtle geometry, modular construction, lattice embedding, magnetization

Media

stainless steel, Corten steel, paper folding, magnetic modules

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 2019 Conference Proceedings

The Obtetrahedrille as a Modular Building Block for 3D Mathematical Art

Tom Verhoeff ${}^1$ and Koos Verhoeff ${}^2$

${}^1$ Dept. Math. & CS, Eindhoven Univ. of Techn., Netherlands, T.Verhoeff@tue.nl ${}^2$ 1927-2018 (this article was in preparation for Bridges 2018)

Abstract

We explain how the obtetrahedrille can serve as a versatile building block for 3D structures. In particular, the obtetrahedrille can be used to construct triangular beams in the body-centered cubic lattice, and square beams in the simple cubic lattice. We present a language to describe structures composed from obtetrahedrilles. We show some sculptures that can be constructed from obtetrahedrilles. Finally, we discuss how to magnetize the obtetrahedrilles to allow simple experimentation.

Figure 1: Swinger by Koos Verhoeff, polished stainless steel, Valkenswaard, Netherlands, 72 cm high (left, middle); obtetrahedrille, brushed stainless steel (right)

Swinger (Fig. 1, left and middle) is a mathematical sculpture designed by Koos Verhoeff. Its beams have an equilateral triangle as cross section and run in the directions of the space diagonals of a (tilted) cube. That makes Swinger a BCC (body-centered cubic) lattice path. Its name is a pun on Swing, a sculpture by Arie Berkulin (Fig. 2, left), constructed from square beams that follow the face diagonals of a cube (Fig. 2, right), making Swing a FCC (face-centered cubic) lattice path.

Swing is a Hamiltonian cycle on a regular tetrahedron. Thus, Swing is a regular constant-torsion polygon [5]: all beam segments have the same length, all angles between adjacent beam segments are the same $(60^{\circ })$ , and all dihedral angles between angle spanning planes of adjacent vertices are the same $(70.53^{\circ })$ . Swinger is not a Hamiltonian cycle on a regular tetrahedron, but it is a regular constant-torsion polygon with joint angles of $70.53^{\circ }$ and dihedral angles of $60^{\circ }$ (dual to Swing). Koos liked Swinger better than Swing, because the former, unlike the latter, has joints with some ‘flush’ beam faces: one beam face is coplanar across the joint.

Obtetrahedrille

Swinger is a Hamiltonian cycle on a tetragonal disphenoid (a tetrahedron with congruent isosceles triangles as faces) of a special kind, viz. having two dihedral angles of $90^{\circ }$ at the two longer edges, and four dihedral

Verhoeff and Verhoeff

Figure 2: Swing by Arie Berkulin, Cor-ten steel, $11\mathrm{m}$ high, Eindhoven, Netherlands, designed in 1969, constructed in 1977 (left; image source: [3]); computer model of Swing inside cube (right)

angles of $60^{\circ }$ at the four shorter edges (see Fig. 1, right, and Fig. 3, left, where the longer edges are aligned with the $Y$ - and $Z$ -axis, and the shorter edges are space diagonals of cubes). Swinger follows those shorter edges. Conway calls this shape an oblate tetrahedrille, or just obtetrahedrille [1, p.294], abbreviated here as OTHD. In [2], Gibb shows how to fold it from a single sheet of A4 paper. The OTHD can be embedded in space such that its vertices have integer coordinates, with two edges of length 2 and four edges of length $\sqrt{3}$ . Unlike the regular tetrahedron, it is a space-filling polyhedron, resulting in what is known as the tetragonal disphenoid honeycomb. This is what makes the OTHD a good candidate for a modular building block.

Figure 3: Obtetrahedrille placed inside four cubes (left); Swinger (middle); variant of Swinger (right)

What is even nicer is that the OTHD can be used to construct two types of beams:

• triangular beams, in the four directions of the cube’s space diagonals (BCC lattice),
• square beams, in the three orthogonal directions (simple cubic, SC lattice).

Fig. 3 (middle) shows how Swinger is built up from OTHDs. On the right, a variant is shown where the cross section of the beams has been rotated through $180^{\circ }$ . Fig. 4 shows various views of Kliekje (‘Leftover’), constructed from square beams, being a regular constant-torsion polygon with bending (turtle turn) angles and dihedral (turtle roll) angles of $90^{\circ }$ . Some other shapes constructed from OTHDs are shown in Fig. 5.

Note that these OTHD beams cannot make turns at arbitrary locations. Their branching points follow a helix around the beam. This is similar to the triangular beams obtained by folding $1:\sqrt{2}$ rhombus strips described in [6]. The faces of the OTHD are half a $1:\sqrt{2}$ rhombus. Thus, the OTHD can construct more varied shapes. For instance, Swinger cannot be constructed from $1:\sqrt{2}$ rhombuses.

The Obtetrahedrille as a Modular Building Block for 3D Mathematical Art

Figure 4: Kliekje by Koos Verhoeff, stainless steel (various views, and its construction from OTHDs)

Figure 5: Trefoil and figure-8 knot from OTHDs (left, right); Rusty Thing by Koos Verhoeff, Corten (middle)

Language to Describe Obtetrahedrille Constructions

It is convenient to have a notation to describe OTHD constructions. We use a language based on 3D turtle geometry [4], where the turtle supports these commands (Fig. 6, left):

• $B$ (Back) the face via which the turtle arrived at the OTHD
• $P$ (Perpendicular) place next OTHD on $P$ -face
• $L$ (Left) place next OTHD on $L$ -face
• $R$ (Right) place next OTHD on $R$ -face
• $H_i$ (History) go back to state at step $i$ (negative for relative to last step)

The turtle travels in the lattice shown in Fig. 6 (middle). This is a sublattice of the FCC lattice, consisting of the edges of truncated octahedrons that fill space. The OTHDs are the Voronoi regions of this lattice.

A repetition of $R$ (or $L$ ) commands describes a triangular beam, where the turtle follows a period-3 helical path (Fig. 6, right a). A repetition of $PL{H}_{-1}R$ describes a square beam (Fig. 6, right b). A rhombic dodecahedron (Fig.7, left) is described by $LRLRRLRLRLRLRLRLRLRLRLRLRLRLRLRLRLRLRLRLRLRLRLRLRLRLRLRLRLRLRLRLRLRLRLRLRLRLRLRLRLRLRLRLRLRLRLRLRLRLRLRLRLRLRLRLRLRLRLRLRLRLRLRLRLRLRLRLRL

Verhoeff and Verhoeff

Figure 6: Turtle commands (left), lattice (middle), triangular and square beams (right $a$ , $b$ )

Magnetizing the Obtetrahedrille

The faces of the OTHD can be magnetized with north and south poles, such that all modules are identical, and can be magnetically joined to fill space. Two faces sharing a longer edge get north poles, and the other two faces, sharing the other longer edge, get south poles. If Fig. 6 (left) the $P$ and $B$ faces have North poles, and the $L$ and $R$ faces have South poles. Fig. 7 (right) shows how everything connects properly.

Figure 7: Rhombic dodecahedron from 24 OTHDs (left); magnetized OTHDs (right)

References

[1] J. Conway, H. Burgiel, and C. Goodman-Strauss. The Symmetries of Things. CRC Press, 2016. https://books.google.nl/books?id=Drj1CwAAQBAJ. [2] W. Gibb. “Paper Patterns 2: Solid Shapes from Metric Paper.” Mathematics in School, vol. 19, no. 3, 1990, pp. 2-4. http://www.jstor.org/stable/30214670. [3] Lempkesfabriek. “Picture of Swing.” http://commons.wikimedia.org/w/index.php?curid=4473555. [4] T. Verhoeff. “3D Turtle Geometry: Artwork, Theory, Program Equivalence and Symmetry.” Int. J. of Arts and Technology, vol. 3, no. 2/3, 2010, pp. 288-319. [5] T. Verhoeff and K. Verhoeff. “Regular 3D Polygonal Circuits of Constant Torsion.” Bridges Conference Proceedings, Banff, Canada, Jul. 26-30, 2009, pp. 223-230. http://archive.bridgesmathart.org/2009/bridges2009-223.. [6] T. Verhoeff and K. Verhoeff. “Folded Strips of Rhombuses and a Plea for the $\sqrt{2}:1$ Rhombus.” Bridges Conference Proceedings, Enschede, the Netherlands, Jul. 27-31, 2013, pp. 71-78. http://archive.bridgesmathart.org/2013/bridges2013-71..