# Bamboozle: A Mathematical Artwork in MetaForum

## [ Facts | Models | Design | Name | References ]

Title Bamboozle Koos Verhoeff and Tom Verhoeff 2012 Eindhoven University of Technology Department of Mathematics & Computer Science MetaForum (Opening hours) 1st floor, elevator hall near the reception 19 mm polished acrylic 51 equilateral triangles (30 cm side length; 15 yellow, 12 red, 12 green, 12 blue) approx. 160 x 160 x 160 cm3 19 January 2013 Glance & Vision, Veldhoven, NL
Bamboozle as it arrived in two pieces on 18 January 2013 Finishing touches Waiting to be installed
Installed Installed (from below) Installed (at an angle)

## Some Facts

Bamboozle consists of 51 equilateral triangles, meeting pairwise at an angle of about 70.5 degrees (arccos 1/3). The four colors correspond to the four orientations of the triangles. There are 15 yellow triangles, and 12 triangles of each red, green, and blue. The smallest cycle involves 10 triangles.

Bamboozle's structure is part of an infinite space-spanning structure with symmetry group I4332 (number 214 in the list of 230 3D space groups). This group contains rotations of order 2 (over 180 degrees) and order 3 (over 120 degrees), and also left-handed screw displacements of order 4 (over 90 degrees), and right-handed screw displacements of order 3. The group has no reflections; the structure is not mirror symmetric. However, the empty space surrounding the triangles has the mirror image as structure (symmetry group I4132); that is, a mirror image of the structure can be nicely woven through the structure itself (see picture below).

Bamboozle with its mirror image interwoven

The spatial structure is known by various names:

• Laves's graph of girth ten
• (10,3)-a
• srs
• K4 crystal
• triamond

This structure is not well known, yet mathematically intriguing, and "ubiquitous" in nature [SH+al, TD]. For instance, it has the strong isotropy property [TS]. The only other structures with this property are diamond (in 3D) and the honeycomb lattice (in 2D; cf. graphene). As paraphrased by Tony Phillips,

[t]he strong isotropy property states that for any two vertices V and W of the crystal, any ordering of the edges adjacent to V and any ordering of the edges adjacent to W, there is a lattice-preserving congruence taking V to W and each V-edge to the similarly ordered W-edge.

## Models

### Slotted triangles

Specially designed triangles slide into each other at the appropriate angle (see figure below).

Slotted triangles and how they slide together at an angle

With a bit of consideration, you can construct a Bamboozle from 51 slotted triangles. Of course, you can also put together other shapes.

Technical details of these triangles:

Side length 30 mm 1.9 mm Nylon (White Strong & Flexible) 3D printed at Shapeways

See [GH].

## Design

In [TV] we describe how Bamboozle was discovered.

Here is a short talk about Bamboozle:

Bamboozle was generated by (see figures below)

1. starting with a single triangle (as abstraction from a trefoil knot made of √2:1 rhombuses),
2. adding five generations of direct neighbors (this is the earliest moment when cycles appear), and
3. omitting, recursively, those triangles that have fewer than two neighbors.

## Name

So, where does the name Bamboozle come from?

Koos Verhoeff gave this name, after we struggled some time to find out the mathematical structure.

bamboozle
verb
fool or cheat; confound or perplex

## References

[GH]
George Hart, The (10,3)-a Network.
[NS]
Norbine Schalij. Ingewikkelde wiskunst fleurt liftruimte op (in Dutch), Cursor on-line, 21 Jan. 2013.
[NT]
Nicole Testerink. Interview met Tom Verhoeff (in Dutch): "De lucht van lijm en houtzaagsel doet mij niet zoveel", Rubriek Mens, Cursor, jaargang 54, nummer 15, p.15 (5 Apr. 2012).
[SH+al]
Stephen T. Hyde, Michael O'Keeffe, and Davide M. Proserpio. "A Short History of an Elusive Yet Ubiquitous Structure in Chemistry, Materials, and Mathematics", Angewandte Chemie International Edition, Volume 47, pages 7996-8000 (2008).
[TD]
Tomonari Dotera. "Quest for the Gyroid Labyrinth: Geometry and Topology in Soft Matter", Journal of the Physical Society of Japan Online, News and Comments (17 Aug. 2012).
[TS]
Toshikazu Sunada. "Crystals That Nature Might Miss Creating", Notices of the AMS, Volume 55, Number 2, pages 208-215 (Feb. 2008).
Also see responses:
• Toshikazu Sunada. Letters to the Editor, Correction: "Crystals That Nature Might Miss Creating", Notices of the AMS, Volume 55, Number 3, page 343 (Mar. 2008).
• Alan H. Schoen. Letters to the Editor, "On the Graph (10-3)-a", Notices of the AMS, Volume 55, Number 6, page 663 (Jun./Jul. 2008).
[TV]
Tom Verhoeff, Koos Verhoeff. "Folded Strips of Rhombuses, and a Plea for the √2:1 Rhombus". In: George W. Hart and Reza Sarhangi (Eds.), Proceedings of Bridges 2013: Mathematics, Music, Art, Architecture, Culture, Tessellations Publishing, ISBN 978-1-938664-06-9, pp.71-78, July 2013.

The Foundation MathArt Koos Verhoeff conserves and manages the artwork of Koos Verhoeff.