# Koos Verhoeff, Tom Verhoeff, Anton Bakker

## Artists

## Statement

The main artist is Koos Verhoeff, retired Full Professor of Computer Science at the Erasumus University Rotterdam, The Netherlands. Koos has designed and constructed mathematical art since the early 1980s. He mainly designs 3D sculptures, constructed in wood, bronze, aluminum, stainless steel, and plastics, often involving mitered beams. He has described himself as more of an explorer and a discoverer than a designer. He wonders and wanders about the abstract world of mathematical structures, looking for forms with intriguing mathematical properties as well as aesthetic appeal. Anton Bakker has rendered various designs by Koos, especially on a larger scale.

## Artworks

Mitered fractal tree (designed late 1980s, first executed in wood), constructed from a beam with a square cross section. When this beam is cut at 45 degrees, the result is a rectangular cut face in the ratio 1:sqrt(2). When this beam is cut twice at 45 degrees, where the cuts are perpendicular, the result is a "roof" consisting of two smaller 1:sqrt(2) rectangular panels. On this roof, two smaller copies of the entire tree are grown. However, because the beam's cross section is a square, the roof can (and is) rotated over 90 degrees (compared to Mitered Fractal Tree I). The branches point in six directions only. The result is highly regular structure. Details can be found in the article by Tom Verhoeff and Koos Verhoeff submitted to Bridges 2012. This bronze cast was constructed under the direction of Anton Bakker.

This sculpture consists of 51 triangular prisms (contracted to "trism"), arranged in one of the eleven chiral (i.e. non-mirror-symmetric) lattices among the 230 space groups. Each trism is connected to three neigbors at an angle of 70.5 degrees. The trisms appear in four orientations (cf. the main diagonals of the cube and the faces of a tetrahedron), represented by the four colors. The shortest cycle consists of 10 trisms. The design arose out of explorations by Koos Verhoeff and Tom Verhoeff of interlocked trefoil knots constructed from mitered triangular beams. To our surprise this interlocking could be extended to a rare lattice. No picture does justice to its 3D beauty ("A sculpture says more than a thousand pictures"). Recently, we discovered that this structure is also known as (10,3)a; its history is presented by George Hart at georgehart.com/rp/10-3.html

Mitered fractal tree (designed late 1980s, first executed in wood), constructed from a beam with a rectangular cross section in the ratio 1:sqrt(2). When this beam is cut at 45 degrees, the result is a square cut face. When this beam is cut twice at 45 degrees, where the cuts are perpendicular, the result is a "roof" consisting of two smaller square panels. On this roof, two smaller copies of the entire tree are grown. No two branches point in the same direction. The result is an awe inspiring organic structure that is both highly structured and chaotic. Details can be found in the article by Tom Verhoeff and Koos Verhoeff submitted to Bridges 2012. This bronze cast was constructed under the direction of Anton Bakker.