Mathematics, geometry is art – shape, structure, form, relations, quantities, measures, order, regularity, rules and exceptions.
Aesthetics – how to calculate the pleasurable, how to measure the appealing – can we avoid mathematics?
Are mathematical constructions science or art? Where is the borderline?
Art is craft – revival of the abstract. From ideas to material objects.
Art is an expression of our knowledge, understanding and awareness.
Description
This string model is a homage to the first material mathematical models used to study ruled surfaces in the 19th century.
Strings represent the rulings, lines entirely contained in the ruled surfaces.
The geometrical structure of the model is a homage to number 4. It represents, as a spatial object, a 4-dimensional cube, called a hypercube or a tesseract, a polytope with 4 edges going out of each vertex.
The outer frame is a 3D cube, Platonic solid with 8 vertices and the inner object is an octahedron, a Platonic solid dual to the cube, with 8 edges.
They are connected with 4 hyperbolic paraboloids, each one containing 1 edge of the cube and 1 edge of the octahedron, 4 vertices of the hypercube.
Hyperbolic paraboloids are 2nd degree ruled surfaces with 2 system of rulings. The strings belong to one system and the 2 edges of the hypercube belong to the other one.
Octahedron is decorated with 2 spirals which pass through 3 vertices of the hypercube.
Materials
The outer frame and the inner octahedron are made from HDF panels cut by a laser cutter. The cube is made from 24 pieces and the octahedron from 5, glued and colored.
The strings are recycled wool.
The spirals are made of one piece of steel wire.
Dimensions:
81x81x81 cm
Description
This star-shaped bamboo sculpture is the second representation of a hypercube, with a similar geometrical frame.
The outer shape is an 8-star, stellated octahedron, named Stella octangula by Johannes Kepler in the early 17th century, whose 8 vertices make a cube with invisible edges.
It was made so, because a square is not a stable figure while a triangle is. The cube is thus constructed as a convex hull of a regular compound of two tetrahedrons, self-dual Platonic solids with 4 vertices.
Geometrically, these two tetrahedrons are congruent and their intersection is an octahedron. Their combined Boolean difference consists of 8 smaller congruent tetrahedrons.
In each of the small tetrahedrons two hyperbolic paraboloids are placed to contain among them all the edges of the tetrahedron.
Small bamboo sticks present one system of rulings of each hyperbolic paraboloid and they are all parallel to the same vertical plane.
Materials
The sculpture is made from plenty bamboo sticks and a bag full of rubber bands.
There are 3 sizes of bamboo sticks, 12 pieces of big (220 cm) and medium (180 cm) each and over 200 pieces of smaller ones (120 cm).
Photo by Marie Bartz
Photo by Masayo Aye
Photo by Masayo Aye
Hypercube On Point (2019)
Iva Kodrnja & Helena Koncul
In this portrayal of a tesseract, the inner cube is replaced by its Platonic dual – an octahedron. Edges of its base square are connected to skew edges of the outer cube by strings which, looking at each pair of connected edges, geometrically define one system of rulings of a hyperbolic paraboloid.
The inspiration for this piece are old string models, used and created for scientific and educational purposes of studying ruled surfaces in the 19th century. Models of this kind can be found in the majority of technical universities throughout the world, and today, with the help of modern technologies such as CNC machines and CAD software, digital production of similar models can be and still is an excellent educational tool.
Materials:
Laser-cut cardboard, wooden tile, glue, spray paint and knitting yarn
Dimensions:
60x56x56 cm
We would like to thank our friends Milena Stavrić and Albert Wiltsche
at Institute for Architecture and Media, TU Graz and Željko Kelković at Department of Engineering Mechanics
at Faculty of Civil Engineering in Zagreb
for their help with the realization of this project.
Octahedron & 8 Parabolae (2019)
Iva Kodrnja & Helena Koncul
Dual to the Steiner construction of conics, by taking two intersecting lines we can construct a number of tangents of a parabola, called a dual parabola.
This simple construction is knitted inside 8 faces of an octahedron.
The octahedron has the ideal amount of complexity and an extraordinary structure consisting of two pyramids.
We would like to thank Željko Kelković at Department of Engineering Mechanics at Faculty of Civil Engineering
for helping us with realization of this project.
