Lobke, and Other Constructions from Conical Segments

Presented at Bridges 201415 August 2014, Seoul, South Korea

Tom VerhoeffEindhoven Univ. of Technology

Dept. of Math. & CS

Koos VerhoeffValkenswaard

The Netherlands

c© 2014, T. Verhoeff @ TUE.NL 1/27 Constructions from Conical Segments

Recent Mitered Designs by Koos Verhoeff

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Lobke (Koos Verhoeff, 1990s)

Fiberglass with polyester resin on a metal mesh (73 cm tall)

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Cones with 90◦ Aperture in Cube

The cones touch (blue lines, on the right)

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3/4 of 90◦ Conical Segments in Cube, Forming a Closed Strip

The segments touch, and connect smoothly (blue edges)

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Six-fold Symmetry

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Self-Intersection

To make a sculpture, the segments must be thickened

Thickening: touching → self-intersection

Self-intersection can be avoided:

• Reduce the aperture of the cones to < 90◦

• Preserve the six-fold symmetry, i.e., the equatorial cut lines

• Preserve smooth connections

• Hence, also reduce the fraction of cone in the segments to < 3/4

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Mathematics Involved

• Cone: tip T , axis `, aperture 2α

• Cut by plane tilted over β

• Angle m1Tm2 is 2γ

• Pythagorean Theorem for right-angled spherical triangles:

cosα = cosβ cos γ

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Conical Segments with Varying Aperture, Sharing Two Edges

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3D Print of Conical Segments Sharing Two Edges

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Reduced Aperture and Fraction

Aperture 86◦ Aperture 60◦

Cone Fraction 0.738 Cone Fraction 1/2

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Variation 1: Vary the Number of Lobes

4 Lobes 8 Lobes 10 Lobes

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Emphatic Self-Intersection

4 Lobes 6 Lobes 8 Lobes

For ceramic 3D prints, self-intersection is necessary

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Ceramic 3D Prints of Self-Intersecting Variants

6 Lobes 10 Lobes

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Variation 2: Vary the Connections between Segments

3 4

1

2

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Problem: Create Properly Closed Smooth Strips

• Using just one type of conical segment

• Parameters of conical segment: aperture 2α, radius r, fraction β

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Describing Strips of Conical Segments

New Turtle Geometry command: CStrip(α, r, β)

CStrip(45◦,1,270◦) CStrip(90◦,1,270◦) CStrip(0,1,270◦)

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Relationship to Connection Types

The following conical segments are congruent:

0. CStrip(α, r, β),

1. CStrip(180◦ − α, r, β),

2. CStrip(180◦+ α, r, β),

3. CStrip(360◦ − α, r, β),

A strip is fully described by α, β, and a sequence of indices

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Find Closed Strips by Trial and Error Elimination

Strip generated by α = 36◦, β = 246± 1◦, sequence (0,1,2,3,2,1)3

Tweak α and/or β to obtain closure

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Mathematica App to Explore Strips of Conical Segments

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Examples of Closed Strips of Conical Segments

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Discrete Approximations of Conical Segments

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Same Shapes with Straight Trapezoidal Segments

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More Examples of Closed Strips of Conical Segments

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Related Work

• Seat of Wisdom and Circle Squared by Vic Pickett

• Bronze Spheric Theme and Model for ‘Spheric Theme’ by NaumGabo

• Snake, Berlin Junction, and other sculptures by Richard Serra

• Borsalino and other sculptures by Henk van Putten, using cylin-drical segments with a square cross section

Also see “LEGOr” Knots by Séquin and Galemmo (Bridges 2014)

• Arabesque XXIX by Robert Langhurst resembles Lobke, but ithas no hole and it is not a developable surface.

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Conclusion

• Explore constructions with congruent conical segments

Two parameters: cone aperture, cone fraction

• Challenge: find properly closed strips

• Describe with Turtle Geometry

• Relationship with mitered constructions

• Relationship with constant torsion paths

• Rotate segments about center line

• Square cross section

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Rotate segment about the center line; square cross section

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