Three Mathematical Sculptures at the Mathematikon

General Information

• Designer: Koos Verhoeff (mathematician, computer scientist, mathematical artist)
• Location: Klaus Tschira Platz, near Im Neuenheimer Feld 231, 69120 Heidelberg, Germany
• Installation Date: 23 October 2015
• Tom Verhoeff, Koos Verhoeff. "Three Mathematical Sculptures for the Mathematikon". Proceedings of Bridges 2016: Mathematics, Music, Art, Architecture, Education, Culture, pp.105-110, 2016.
  [ Slides ]

Lobke

• Mathemical Inspiration: The octahedron (the Platonic solid with 8 triangular faces), via an Euler cycle
• Origin of Name: Lobke is an archaic/affective diminuitive of the Dutch word "lob", which means "lobe"; the sculpture has six "lobes"
• Height: 3.5 m
• Mass: approx. 1200 kg
• Material: Stainless Steel (AISI 316), polished and glass-bead blasted
• Parts: Six conical segments
• Symmetry group: *223 (Orbifold notation); 12 symmetries; achiral
  • order-3 (120-degree) rotational symmetry about vertical axis
  • 60-degree rotoreflective symmetry about vertical axis
  • mirror symmetric in three vertical planes
• Mathematical Construction:

  Tom Verhoeff, Koos Verhoeff. "Lobke, and Other Constructions from Conical Segments". In: Gary Greenfield, George Hart and Reza Sarhangi (Eds.), Proceedings of Bridges 2014: Mathematics, Music, Art, Architecture, Culture, Tessellations Publishing, ISBN 978-1-938664-11-3, pp.309-316, August 2014.
  [ Slides | Additional material ]

Balancing Act

• Mathemical Inspiration: The figure-eight knot, via an embedding in the body-centered cubic lattice
• Origin of Name: All beams have been rotated along their longitudinal axis, so that the sculpture will balance on a single beam when it stands freely
• Diameter: 2.4 m
• Mass: approx. 350 kg
• Material: Stainless Steel (AISI 316), polished
• Parts: 16 beams, triangular cross section (triangle side length 40 cm), connected by regular miter joints

  All joint angles equal ≈109.5 degrees; all torsion angles equal 60 degrees (plus or minus); total torsion is zero

  The vertices have integer coordinates; beam directions are the main diagonals of the cube (body-centered cubic lattice)

• Symmetry group (of center line): 2x (Orbifold notation); 4 symmetries; achiral
  • order-2 (180-degree) rotational symmetry
  • ±90-degree rotoreflective symmetry
• Mathematical Construction:

  Tom Verhoeff. "3D Turtle Geometry: Artwork, Theory, Program Equivalence and Symmetry". Int. J. of Arts and Technology, 3(2/3):288-319 (2010).
  [ Full Text in TU/e Library (campus only) | Slides ]

Hamilton Cycle on Football

• Mathematical Inspriration: The truncated icosahedron, via a Hamiltonian cycle
• Origin of Name: The football (Am. Eng. soccer ball) is traditionally covered with the faces of a truncated icosahedron, which has 32 faces (12 pentagons and 20 hexagons), 30 vertices, and 90 edges; it admits several Hamiltonian cycles; this is the most symmetric one
• Diameter: 2.4 m
• Mass: approx. 350 kg
• Material: Stainless Steel (AISI 316), glass-bead blasted
• Parts: 30 beams, square cross section (square side length 15 cm), connected by regular miter joints
• Symmetry group (of center line): 223 (Orbifold notation); 6 symmetries; chiral
  • order-3 (120-degree) rotational symmetry about vertical axis
  • 60-degree rotoreflective symmetry about vertical axis

Further References

• About miter joints:

  Tom Verhoeff, Koos Verhoeff. "The Mathematics of Mitering and Its Artful Application". Appeared in Bridges Leeuwarden: Mathematical Connections in Art, Music, and Science, Proceedings of the Eleventh Annual Bridges Conference, in Leeuwarden, The Netherlands, pp.225-234, July 2008.
  Also see related Wolfram Mathematica Demonstrations (Mathematica or free CDF Player required):

  • Miter Joint and Fold Joint
  • Mitering a Closed 3D Path
• Construction company: Roestvrijstaalindustrie Geton (Geton Stainless Steel Industry), Veldhoven, Netherlands
• Funding: Klaus Tschira Foundation