Presentation on theme: "4D Polytopes and 3D Models of Them"— Presentation transcript:
1 4D Polytopes and 3D Models of Them Florida 19994D Polytopes and 3D Models of ThemGeorge W. HartStony Brook University
2 Goals of This Talk Expand your thinking. Florida 1999Goals of This TalkExpand your thinking.Visualization of four- and higher-dimensional objects.Show Rapid Prototyping of complex structures.Note: Some Material and images adapted from Carlo Sequin
3 What is the 4th Dimension ? Florida 1999What is the 4th Dimension ?Some people think: “it does not really exist” “it’s just a philosophical notion” “it is ‘TIME’ ”But, a geometric fourth dimension is as useful and as real as 2D or 3D.
4 Higher-dimensional Spaces Florida 1999Higher-dimensional SpacesCoordinate Approach:A point (x, y, z) has 3 dimensions.n-dimensional point: (d1, d2, d3, d4, ..., dn).Axiomatic Approach:Definition, theorem, proof...Descriptive Geometry Approach:Compass, straightedge, two sheets of paper.
5 What Is a Regular Polytope? Florida 1999What Is a Regular Polytope?“Polytope” is the generalization of the terms “Polygon” (2D), “Polyhedron” (3D), … to arbitrary dimensions.“Regular” means: All the vertices, edges, faces… are equivalent.Assume convexity for now.Examples in 2D: Regular n-gons:
6 Regular Convex Polytopes in 3D Florida 1999Regular Convex Polytopes in 3DThe Platonic Solids:There are only 5. Why ? …
7 Why Only 5 Platonic Solids ? Florida 1999Why Only 5 Platonic Solids ?Try to build all possible ones:from triangles: 3, 4, or 5 around a corner;from squares: only 3 around a corner;from pentagons: only 3 around a corner;from hexagons: floor tiling, does not close.higher n-gons: do not fit around vertex without undulations (not convex)
8 Constructing a (d+1)-D Polytope Florida 1999Constructing a (d+1)-D PolytopeAngle-deficit = 90°2D3DForcing closure:?3D4Dcreates a 3D cornercreates a 4D corner
9 Florida 1999“Seeing a Polytope”Real “planes”, “lines”, “points”, “spheres”, …, do not exist physically.We understand their properties and relationships as ideal mental models.Good projections are very useful. Our visual input is only 2D, but we understand as 3D via mental construction in the brain.You are able to “see” things that don't really exist in physical 3-space, because you “understand” projections into 2-space, and you form a mental model.We will use this to visualize 4D Polytopes.
10 Florida 1999ProjectionsSet the coordinate values of all unwanted dimensions to zero, e.g., drop z, retain x,y, and you get a orthogonal projection along the z-axis. i.e., a 2D shadow.Linear algebra allows arbitrary direction.Alternatively, use a perspective projection: rays of light form cone to eye.Can add other depth queues: width of lines, color, fuzziness, contrast (fog) ...
11 Wire Frame Projections Florida 1999Wire Frame ProjectionsShadow of a solid object is mostly a blob.Better to use wire frame, so we can see components.
14 3D Objects Need Physical Edges Florida 19993D Objects Need Physical EdgesOptions:Round dowels (balls and stick)Profiled edges – edge flanges convey a sense of the attached faceFlat tiles for faces – with holes to make structure see-through.
16 How Do We Find All 4D Polytopes? Florida 1999How Do We Find All 4D Polytopes?Sum of dihedral angles around each edge must be less than 360 degrees.Use the Platonic solids as “cells”Tetrahedron: 70.5°Octahedron: 109.5°Cube: 90°Dodecahedron: 116.5°Icosahedron: 138.2°.
17 All Regular Convex 4D Polytopes Florida 1999All Regular Convex 4D PolytopesUsing Tetrahedra (70.5°):3 around an edge (211.5°) (5 cells) Simplex4 around an edge (282.0°) (16 cells) Cross polytope5 around an edge (352.5°) (600 cells)Using Cubes (90°):3 around an edge (270.0°) (8 cells) HypercubeUsing Octahedra (109.5°):3 around an edge (328.5°) (24 cells)Using Dodecahedra (116.5°):3 around an edge (349.5°) (120 cells)Using Icosahedra (138.2°): none: angle too large.
18 5-Cell or 4D Simplex 5 cells, 10 faces, 10 edges, 5 vertices. Florida 19995-Cell or 4D Simplex5 cells, 10 faces, 10 edges, 5 vertices.Carlo SequinCan make with Zometool also
19 16-Cell or “4D Cross Polytope” Florida 199916-Cell or “4D Cross Polytope”16 cells, 32 faces, 24 edges, 8 vertices.
20 4D Hypercube or “Tessaract” Florida 19994D Hypercube or “Tessaract”8 cells, 24 faces, 32 edges, 16 vertices.
36 120-Cell “Net” with stack of 10 dodecahedra George Olshevski Florida 1999120-Cell “Net”with stack of 10 dodecahedraGeorge Olshevski
37 600-Cell -- 2D projection Oss, 1901 Florida 1999600-Cell D projectionTotal: tetra-cells, faces, edges, vertices.At each Vertex: tetra-cells, faces, edges.Oss, 1901Frontispiece of Coxeter’s book “Regular Polytopes,”
38 Cross-eye Stereo Picture by Tony Smith Florida 1999600-CellCross-eye Stereo Picture by Tony Smith
39 Florida 1999600-CellDual of 120 cell.600 cells, faces, edges, vertices.Cell-first parallel projection, shows less than half of the edges.Can make with Zometool
40 Florida 1999600-CellStraw model by David Richter
41 Slices through the 600-Cell Florida 1999Slices through the 600-CellGordon KindlmannAt each Vertex: 20 tetra-cells, 30 faces, 12 edges.
42 History 3D Models of 4D Polytopes Florida 1999History 3D Models of 4D PolytopesLudwig Schlafli discovered them in Worked algebraically, no pictures in his paper. Partly published in 1858 and 1862 (translation by Cayley) but not appreciated.Many independent rediscoveries and models.
43 Stringham (1880) First to rediscover all six Florida 1999Stringham (1880)First to rediscover all sixHis paper shows cardboard models of layers3 layers of 120-cell(45 dodecahedra)
44 Victor Schlegel (1880’s) Five regular polytopes Florida 1999Victor Schlegel (1880’s)Invented “Schlegel Diagram”3D 2D perspective transf.Used analogous 4D 3Dprojection in educationalmodels.Built wire and thread models.Advertised and sold models via commercial catalogs: Dyck (1892) and Schilling (1911).Some stored at Smithsonian.Five regular polytopes
45 Sommerville’s Description of Models Florida 1999Sommerville’s Description of Models“In each case, the external boundary of the projection represents one of the solid boundaries of the figure. Thus the 600-cell, which is the figure bounded by 600 congruent regular tetrahedra, is represented by a tetrahedron divided into 599 other tetrahedra … At the center of the model there is a tetrahedron, and surrounding this are successive zones of tetrahedra. The boundaries of these zones are more or less complicated polyhedral forms, cardboard models of which, constructed after Schlegel's drawings, are also to be obtained by the same firm.”
46 Cardboard Models of 120-Cell Florida 1999Cardboard Models of 120-CellFrom Walther Dyck’s 1892 Math and Physics Catalog
47 Paul S. Donchian’s Wire Models Florida 1999Paul S. Donchian’s Wire Models1930’sRug Salesman with“visions”Wires doubled to show how front overlays backWidely displayedCurrently on view at the Franklin Institute
48 Florida 1999Zometool1970 Steve Baer designed and produced "Zometool" for architectural modelingMarc Pelletier discovered (when 17 years old) that the lengths and directions allowed by this kit permit the construction of accurate models of the 120-cell and related polytopes.The kit went out of production however, until redesigned in plastic in 1992.
50 Uniform 4D Polytopes Analogous to the 13 Archimedean Solids Florida 1999Uniform 4D PolytopesAnalogous to the 13 Archimedean SolidsAllow more than one type of cellAll vertices equivalentAlicia Boole Stott listed many in 1910Now over 8000 knownCataloged by George Olshevski and Jonathan Bowers
59 Big Polytope Zome Model Florida 1999Big Polytope Zome ModelSteve Rogers
60 Florida 199948 Truncated CubesPoorly designed FDM model
61 Prism on a Snub Cube – “Net” Florida 1999Prism on a Snub Cube – “Net”George Olshevski
62 Duo-Prisms - “Nets” Andrew Weimholt George Olshevski Andrew Weimholt Florida 1999Duo-Prisms - “Nets”Andrew WeimholtGeorge OlshevskiAndrew WeimholtRobert Webb
63 Grand Antiprism “Net” with stack of 10 pentagonal antiprisms Florida 1999Grand Antiprism “Net”with stack of 10 pentagonal antiprismsGeorge Olshevski
64 Non-Convex Polytopes Components may pass through each other Florida 1999Non-Convex PolytopesComponents may passthrough each otherSlices may be usefulfor visualizationSlices may bedisconnectedJonathan Bowers
65 Beyond 4 Dimensions … What happens in higher dimensions ? Florida 1999Beyond 4 Dimensions …What happens in higher dimensions ?How many regular polytopes are there in 5, 6, 7, … dimensions ?Only three regular types:Hypercubes — e.g., cubeSimplexes — e.g., tetrahedronCross polytope — e.g., octahedron
66 Hypercubes A.k.a. “Measure Polytope” Florida 1999HypercubesA.k.a. “Measure Polytope”Perpendicular extrusion in nth direction:1D D D D
68 Florida 1999Simplex SeriesConnect all the dots among n+1 equally spaced vertices: (Put next one “above” center of gravity) D D DThis series also goes on indefinitely.
69 7D Simplex A warped cube avoids intersecting diagonals. Florida 19997D SimplexA warped cube avoids intersecting diagonals.Up to 6D can be constructed with Zometool.Open problem: 7D constructible with Zometool?
70 A square frame for every pair of axes Florida 1999Cross Polytope SeriesPlace vertex in + and – direction on each axis, a unit-distance away from origin.Connect all vertex pairs that lie on different axes. 1D D D DA square frame for every pair of axes6 square frames = 24 edges
71 6D Cross Polytope 12 vertices suggests using icosahedron Florida 19996D Cross Polytope12 vertices suggests using icosahedronCan do with Zometool.
73 Florida 1999Some ReferencesLudwig Schläfli: “Theorie der vielfachen Kontinuität,” 1858, (published in 1901).H. S. M. Coxeter: “Regular Polytopes,” 1963, (Dover reprint).Tom Banchoff, Beyond the Third Dimension, 1990.G.W. Hart, “4D Polytope Projection Models by 3D Printing” to appear in Hyperspace.Carlo Sequin, “3D Visualization Models of the Regular Polytopes…”, Bridges 2002.
74 Florida 1999PuzzleWhich of these shapes can / cannot be folded into a 4D hypercube?Hint: Hold the red cube still and fold the others around it.Scott Kim
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