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Home > Carlo Séquin :: Mathematical Images

Most viewed  Carlo Séquin :: Mathematical Images 
"Hilbert Cube 512"4296 views"Hilbert Cube" is a spacefilling recursive curve in 3 dimensions in analogy to the famous Hilbert curve in the plane. Special care has been taken never to place more than 3 coplanar line segments in sequence. At the largest recursion step the geometry has been slightly altered so as to obtain a closed loop. In the proper parallel projection one can see that the 2 halves of this sculpture are connected by only 2 tube segments. This piece of art gives me the association of an abstract, constructivist model of the human brain. See more views of the
"Hilbert Cube 512".  Carlo Sequin


"Poincare FishDish," by Carlo Sequin, University of California, Berkeley3146 viewsA tiling with regular heptagons does not fit into the Euclidean plane, since 3 times the dihedral angle of the heptagon exceeds 360 degrees. But if we are willing to introduce a progressive scale factor, then the whole hyperbolic plane can be fit into the Poincaré disc. Here is a visualization of a {7,3} tessellation where 3 heptagons join at every vertex, using a tiling motif inspired by the famous Dutch artist M.C. Escher. Each heptagon is cut into 7 identical pizza slices with irregular boundaries in the shape of fish that properly interlock with one another. See more tiling patterns on the Poincare disc.  Carlo Sequin


"Lizard Tetrus," by Carlo Sequin, University of California, Berkeley3029 views24 Lizard tiles, inspired by one of the many planar tilings by M.C. Escher, are mapped around a rounded tetrahedral frame of genus 3. This tiling is a contorted version of the pattern of 24 heptagons displayed on the surface of the marble sculpture "Eightfold Way" by Helaman Ferguson. That sculpture celebrates Felix Kelin's famous "Quartic Curve" which achieves the maximal symmetry of 168 automorphisms possible on a genus3 surface. Read more about patterns on the Tetrus surface.. Thanks to Pushkar Joshi and Allen Lee for their help with mapping Escher tiles onto the tetrus.  Carlo Sequin


"Knot divided" (snow sculpture), by Carlo Sequin (University of California, Bekeley), Stan Wagon (Team Captain), John Sullivan, Dan Schwalbe, and Rich Seeley2535 viewsCan a DIVIDED KNOT be NOT DIVIDED? When carving this sculpture out of a 10x10x12 foot block of hard compacted snow, we started with the simplest possible knot: the overhand knot, also known as the trefoil knot. We then split lengthwise the whole ribbon forming the three big loops. But there is a twist that may lead to surprises: The original knotted strand was actually a triply twisted Moebius band! Thus the question: Does our cut separate the structure into two pieces, or does it form a single, highly knotted twisted strand? Read more about this snow sculpture.  Carlo Sequin


"The Regular Hendecachoron," computer model by Carlo Sequin, University of California, Berkeley.1965 viewsThis hendecachoron (a literal translation of "11cell" into Greek) is a regular, selfdual, 4dimensional polytope composed from eleven nonorientable, selfintersecting hemiicosahedra. This object also has 11 vertices (shown as spheres), 55 edges (shown as thin cylindrical beams), and 55 triangular faces (shown as cutout frames). Different colors indicate triangles belonging to different cells. This intriguing object of high combinatorial symmetry was discovered in 1976 by Branko Grünbaum and later rediscovered and analyzed from a group theoretic point of view by geometer H.S.M. Coxeter. Freeman Dyson, the renowned physicist, was also much intrigued by this shape and remarked in an essay: "Plato would have been delighted to know about it." The hendecachoron has 660 combinatorial automorphisms, but these can only show themselves as observable geometric symmetries in 10dimensional space or higher. In this image, the model of the hendecachoron is shown with a background of a deep space photo of our universe, to raise the capricious question, whether this 10dimensional object might serve as a building block for the 10dimensional universe that some stringtheorists have been postulating.
A more detailed description and visualization of the 11Cell, describing its construction in bottomup as well as in top down ways, can be found in a paper by Sequin and Lanier: “Hyperseeing the Regular Hendecachoron”. There are additional images and VRML models for interactive inspection here.  Carlo Sequin


"Recursive Figure8 Knot" by Carlo Sequin, University of California, Berkeley1255 viewsIn some depictions of a mathematical knot, some of the meshes formed between the crisscrossing strands resemble the overall outline shape of the whole knot. It is then possible to fit a reduced copy of the knot into every one of these meshes and reconnect the strands so as to obtain again a mathematical knot consisting of a single closed strand. Then this process can be continued recursively resulting in a selfsimilar pattern. This general process was applied to the 4crossing Figure8 knot. But rather than performing this process in a drawing plane as outlined above, subsequent generations of reduced knot instances were placed in planes that are roughly perpendicular to one another, resulting in a truly 3dimensional sculpture.  Carlo Sequin


"Arabic Icosahedron" by Carlo Sequin, University of California, Berkeley1252 viewsMoorish patterns found in the Alhambra often depict lattices of interlocking knots. Here such a pattern composed of interlocking trefoil knots has been wrapped around an icosahedron. Each of the 20 faces is replaced with a trefoil knot, which interlocks along the triangle edges with three adjacent trefoils. The exact nature of the linking between adjacent trefoils leaves some freedom to the designer: In the simplest case two adjacent trefoils interlock with just one lobe each. In the "Arabic Icosahedron" they are linked with two lobes each, resulting in a much tighter meshing.  Carlo Sequin


"Birds in the Sky" by Carlo Sequin, University of California, Berkeley1089 viewsThe surface of a sphere is divided into 24 identical regions with the same symmetries as an oriented octahedron. These tiles have birdlike shapes inspired by the work of M.C. Escher. Half the tiles are yellow and have a relief pattern that clearly identifies them as birds. The other 12 tiles are blue without a special relief pattern; they can thus be seen as either the shadows or profiles of birds, or alternatively as blue sky background.  Carlo Sequin


"125 tetrahedra in 25 projected 5cells," (I) by Carlo Séquin, University of California, Berkeley378 viewsThe 5cell (the fourdimensional simplex) is composed of five tetrahedra. It has been projected into 3D Euclidean space so as to form a threesided bipyramid. Twentyfive of those cells again outline the same projection of a 5cell, with five cells at the vertices and two cells along each of the ten edges. The whole sculpture comprises 125 small tetrahedra, to celebrate the 125th birthday of the AMS. It was designed with SLIDE at U.C. Berkeley, and fabricated in 571 layers, each 0.01 inch thick, on a Fused Deposition Modeling machine.  Carlo Séquin
This view (I) is featured on a printed AMS 125th anniversary poster, free upon request.


"125 tetrahedra in 25 projected 5cells," (II) by Carlo Séquin, University of California, Berkeley206 viewsThe 5cell (the fourdimensional simplex) is composed of five tetrahedra. It has been projected into 3D Euclidean space so as to form a threesided bipyramid. Twentyfive of those cells again outline the same projection of a 5cell, with five cells at the vertices and two cells along each of the ten edges. The whole sculpture comprises 125 small tetrahedra, to celebrate the 125th birthday of the AMS. It was designed with SLIDE at U.C. Berkeley, and fabricated in 571 layers, each 0.01 inch thick, on a Fused Deposition Modeling machine.  Carlo Séquin
This work is featured on a printed AMS 125th anniversary poster, free upon request.


"125 tetrahedra in 25 projected 5cells," (III) by Carlo Séquin, University of California, Berkeley140 viewsThe 5cell (the fourdimensional simplex) is composed of five tetrahedra. It has been projected into 3D Euclidean space so as to form a threesided bipyramid. Twentyfive of those cells again outline the same projection of a 5cell, with five cells at the vertices and two cells along each of the ten edges. The whole sculpture comprises 125 small tetrahedra, to celebrate the 125th birthday of the AMS. It was designed with SLIDE at U.C. Berkeley, and fabricated in 571 layers, each 0.01 inch thick, on a Fused Deposition Modeling machine.  Carlo Séquin
This work is featured on a printed AMS 125th anniversary poster, free upon request.


"125 tetrahedra in 25 projected 5cells," (IV) by Carlo Séquin, University of California, Berkeley135 viewsThe 5cell (the fourdimensional simplex) is composed of five tetrahedra. It has been projected into 3D Euclidean space so as to form a threesided bipyramid. Twentyfive of those cells again outline the same projection of a 5cell, with five cells at the vertices and two cells along each of the ten edges. The whole sculpture comprises 125 small tetrahedra, to celebrate the 125th birthday of the AMS. It was designed with SLIDE at U.C. Berkeley, and fabricated in 571 layers, each 0.01 inch thick, on a Fused Deposition Modeling machine.  Carlo Séquin
This work is featured on a printed AMS 125th anniversary poster, free upon request.


"125 tetrahedra in 25 projected 5cells," (V) by Carlo Séquin, University of California, Berkeley135 viewsThe 5cell (the fourdimensional simplex) is composed of five tetrahedra. It has been projected into 3D Euclidean space so as to form a threesided bipyramid. Twentyfive of those cells again outline the same projection of a 5cell, with five cells at the vertices and two cells along each of the ten edges. The whole sculpture comprises 125 small tetrahedra, to celebrate the 125th birthday of the AMS. It was designed with SLIDE at U.C. Berkeley, and fabricated in 571 layers, each 0.01 inch thick, on a Fused Deposition Modeling machine.  Carlo Séquin
This work is featured on a printed AMS 125th anniversary poster, free upon request.


"125 tetrahedra in 25 projected 5cells," (VII) by Carlo Séquin, University of California, Berkeley111 viewsThe 5cell (the fourdimensional simplex) is composed of five tetrahedra. It has been projected into 3D Euclidean space so as to form a threesided bipyramid. Twentyfive of those cells again outline the same projection of a 5cell, with five cells at the vertices and two cells along each of the ten edges. The whole sculpture comprises 125 small tetrahedra, to celebrate the 125th birthday of the AMS. It was designed with SLIDE at U.C. Berkeley, and fabricated in 571 layers, each 0.01 inch thick, on a Fused Deposition Modeling machine.  Carlo Séquin
This work is featured on a printed AMS 125th anniversary poster, free upon request.


"125 tetrahedra in 25 projected 5cells," (VI) by Carlo Séquin, University of California, Berkeley104 viewsThe 5cell (the fourdimensional simplex) is composed of five tetrahedra. It has been projected into 3D Euclidean space so as to form a threesided bipyramid. Twentyfive of those cells again outline the same projection of a 5cell, with five cells at the vertices and two cells along each of the ten edges. The whole sculpture comprises 125 small tetrahedra, to celebrate the 125th birthday of the AMS. It was designed with SLIDE at U.C. Berkeley, and fabricated in 571 layers, each 0.01 inch thick, on a Fused Deposition Modeling machine.  Carlo Séquin
This work is featured on a printed AMS 125th anniversary poster, free upon request.



