## Geometrikum - geometric exhibits to explore on the 3rd floor of Science Park 2 ###### Developable surface
From Wikipedia

Sample 1: Möbius strip - 3D
Printable pattern page 1 (pdf/ svg), page 2 (pdf/ svg). Blue is mountain folding, red is valley folding.

Sample 2: Stanford bunny A Python script for unfolding triangular meshes in order to create paperfold models at Github

construction manual:
Sheet size 30x30 cm, with numbers: SVG
Sheet size 30x30 cm, without numbers: SVG sheet 1, 2, 3, 4,
Sheet size A4, without numbers: PDF sheet 1, 2, 3, 4,

↑ Top  ###### Villarceau circles

In geometry, Villarceau circles are a pair of circles produced by cutting a torus obliquely through the center at a special angle (from Wikipedia).

Construction period: May 2016 - July 2016, more details here.

↑ Top ###### Hyperboloid of revolution

A hyperboloid of revolution of one sheet. The strings are straight lines. For any point on the surface, there are two straight lines lying entirely on the surface which pass through the point. This illustrates the doubly ruled nature of this surface. From Wikipedia

Construction period: Feb. 2015 - March 2015, more details here.

↑ Top  ###### Menger Sponge and vice versa, card cubes

In mathematics, the Menger sponge is a fractal curve. It is a three-dimensional generalization of the Cantor set and Sierpinski carpet. It was first described by Karl Menger in 1926, in his studies of the concept of topological dimension, see Wikipedia.

Instructions from the institute for figuring. Card cube instructions.

Construction period: May 2014 - July 2014. No glue was used! More details here. ↑ Top ###### Grid of Klein bottle made of cable ties in 2017   ↑ Top ###### Tangent Developable made of a plexiglass tube and strings

2018

The tangent developable of a space curve γ(t) is a developable surface formed by the union of the tangent lines to the curve. From Wikipedia

↑ Top ###### Plücker's conoid

In geometry, Plückers conoid is a ruled surface named after the German mathematician Julius Plücker. From Wikipedia

↑ Top ###### Lorenz manifold, crocheted

Weather forecasting is an inaccurate science. Especially the patterns of clouds and their effects are a mystery. One of the first scientists who ventured into the mysteries of weather was the meteorologist Edward Lorenz. In 1963 Lorenz worked on a calculation for weather forecasting, more precisely he developed a simplified mathematical model for atmospheric convection. From these calculations he received the Lorenz Attractor, from which the Lorenz Manifold was developed in 2002. In 2002 the mathematician Hinke Osinga discovered that the computer model of Lorenz Manifold looks like a crochet script. Osinga did not hesitate for long and crocheted the first Lorenz Manifold. The model of the geometricum consists like the original model of 47 rows and 21.989 stitches. In total more than 1 km of yarn was used to crochet the model.  ↑ Top ###### Hyperbolic plane

2013, more information of the discoverer of hyperbolic crochet Daina Taimina on the homepage of the Crochet Coral Reef project.

Crocheted hyperbolic models usually consist of double crochets, single crochets, chain stitches and slip stitches. They are crocheted in rounds and every \$n\$-th stitch will be doubled. The smaller \$n\$ is, the more crimped the hyperbolic plane becomes.  ↑ Top ###### Tesseract - Hypercube

A hypercube is an n-dimensional analogy to a cube or a square. A 4D-hypercube is also referred to as Tesseract.
In order to recieve a 3D-projection of the Tesseract, one will start with it's 3D-net (shown on the left picture also called Dali cross). By cutting and folding it can be turned into the second figure (right), before reaching the final representation (top). Over all there are 261 possibilities to display the 3D-net of a 4D-Hypercube.  ↑ Top ###### Truncated icosahedron

In geometry, the truncated icosahedron is an Archimedean solid, one of 13 convex isogonal nonprismatic solids whose faces are two or more types of regular polygons. This geometry is associated with footballs typically patterned with white hexagons and black pentagons. From Wikipedia   ###### Sierpinski Tetrahedron

A Sierpinski Tetrahedron was built of straws during the long night of science 2014. ↑ Top ###### Stanford bunny

We built the Stanford bunny out of Lego pieces for the long night of science 2016. ↑ Top ###### Oloid

An oloid is a three-dimensional curved geometric object that was discovered by Paul Schatz in 1929. It is the convex hull of a skeletal frame made by placing two linked congruent circles in perpendicular planes, so that the center of each circle lies on the other circle. The distance between the circle centers equals the radius of the circles. One third of each circle's perimeter lies inside the convex hull, so the same shape may be also formed as the convex hull of the two remaining circular arcs each spanning an angle of 4π/3. From Wikipedia

Construction period: autumn 2017.

↑ Top ###### 3-D Hilbert curve

A Hilbert curve (also known as a Hilbert space-filling curve) is a continuous fractal space-filling curve first described by the German mathematician David Hilbert in 1891, as a variant of the space-filling Peano curves discovered by Giuseppe Peano in 1890. From Wikipedia

↑ Top ###### Trefoil knot

In knot theory, a branch of mathematics, the trefoil knot is the simplest example of a nontrivial knot. The trefoil can be obtained by joining together the two loose ends of a common overhand knot, resulting in a knotted loop. As the simplest knot, the trefoil is fundamental to the study of mathematical knot theory. . From Wikipedia

Construction plan

↑ Top ###### 120-cell or hyperdodecahedron

In geometry, the 120-cell is the convex regular 4-polytope with Schläfli symbol {5,3,3}.
The boundary of the 120-cell is composed of 120 dodecahedral cells with 4 meeting at each vertex. It can be thought of as the 4-dimensional analog of the regular dodecahedron. Just as a dodecahedron can be built up as a model with 12 pentagons, 3 around each vertex, the dodecaplex can be built up from 120 dodecahedra, with 3 around each edge. From Wikipedia

↑ Top ###### 24-cell

The 24-cell is a convex regular fourdimensional polytope, which was first described by the swiss mathematician Ludwig Schläfli in the mid-19th century. It is also known as icositetrachoron, octaplex, icosatetrahedroid, octacube, hyperdiamond or polyoctahedron, being constructed of octahedral cells.

Its boundary consists of 24 (regular) octahedron cells, six of which meet at each vertex and three at each edge. The body contains, in addition to the 24 cells, 96 triangular planes, 96 edges and 24 corners.

The present model was built as part of a seminar at the Institute of Applied Geometry of metal and fishing line.

↑ Top ###### Stellated Rhombicosidodecahedron
From Wikipedia

no glue was used!

↑ Top ###### The Schwarz Lantern

2018, The Schwarz lantern is a pathological example of the difficulty of approximating a smooth curved surface with a polyhedron. From Wikipedia

↑ Top  ###### Soma cube made of card cubes

The Soma cube was invented in 1936 by Piet Hein during a lecture on quantum mechanics conducted by Werner Heisenberg. Its primary function is as an dissection puzzle with the goal of constructing a 3x3x3 cube from seven smaller, individual parts.

There are 240 distinct solutions of the Soma cube puzzle, except for when the cube is rotated or flipped.

Card cube instructions.

↑ Top

###### Paper works

including a 12-unit stellated octahedron and a 30-unit stellated icosahedron made of Sonobe modules, a truncated icosahedron, furthermore, a Bascetta-Stern.

No glue was used! ↑ Top ###### Hexastix

You see four hexagonal tubes which are hollow. The shape of the cavity is a "rhombic dodecahedron", a geometric solid bounded by twelve rhombuses. From George Hart.

↑ Top ###### Match Cube - Match Circle

2014, 2017, no glue was used!

Match Circle - they see me rolling

↑ Top  Johannes Kepler Universität Linz, Institut für Angewandte Geometrie, Altenberger Str.69, 4040 Linz