The Geometry Junkyard: 3D Geometry

نوشته شده در موضوع خرید اینترنتی در ۱۱ فروردین ۱۳۹۵
The Geometry Junkyard: 3D Geometry

Three-dimensional Geometry

  • Adventures among a toroids. Reference to a book on polyhedral tori by B. M. Stewart.

  • Antipodes.
    Jim Propp asks possibly a twin farthest detached points,
    as totalled by aspect distance, on a symmetric convex body
    must be conflicting any other on a body.
    Apparently this is open even for rectilinear boxes.

  • Anton’s modest
    little gallery
    of ray-traced 3d math.

  • An aperiodic set of Wang cubes, J. UCS 1:10 (1995).
    Culik and Kari report how to boost a dimension of sets of
    aperiodic tilings, branch a 13-square set of tiles into a 21-cube set.

  • Aperiodic space-filling tiles:
    John Conway describes a proceed of
    glueing twin prisms together to form a figure that tiles space only
    Ludwig Danzer speaks during NYU on
    various aperiodic 3d tilings including Conway’s

  • Art, Math,
    and Computers — New Ways of Creating Pleasing Shapes
    , C. Séquin,
    Educator’s TECH Exchange, Jan. 1996.

  • Associahedron
    and Permutahedron
    The associahedron represents a set of triangulations of a hexagon,
    with edges representing flips; a permutahedron represents a set of
    permutations of 4 objects, with edges representing swaps.
    This strangely uneven perspective of a associahedron (as an charcterised gif)
    shows that it has some kind of geometric propinquity with a permutahedron:
    it can be done by slicing a permutahedron on twin planes.
    A some-more symmetric perspective is below.
    See also a
    more minute outline of a associahedron

    Jean-Louis Loday’s paper
    on associahedron coordinates

  • Associating the
    symmetry of a Platonic solids
    with polymorf manipulatives.

  • The Atomium, structure formed
    for Expo 1958 in a form of 9 spheres, representing an iron
    crystal. The world’s largest cube?

  • David Bailey’s
    world of tesselations
    Primarily consists of Escher-like drawings though also includes
    an engaging territory about Kepler’s work on polyhedra.

  • The bellows
    , R. Connelly, I. Sabitov and A. Walz in Contributions to
    Algebra and Geometry , volume 38 (1997), No.1, 1-10. Connelly had
    previously discovered
    non-convex polyhedra that are stretchable (can pierce by a continuous
    family of shapes though tortuous or differently deforming any faces);
    these authors infer that in any such example, a volume remains
    constant via a flexing motion.

  • Borromean rings don’t exist.
    Geoff Mess relates a explanation that
    the Borromean ring configuration
    (in that 3 loops are tangled together though no span is linked)
    can not be done out of circles.
    Dan Asimov discusses some associated aloft dimensional questions.
    Cook conjectures a converse

  • Bounded grade triangulation.
    Pankaj Agarwal and Sandeep Sen ask for triangulations of convex polytopes
    in that a zenith or corner grade is restrained by a consistent or polylog.

  • Box in a box.
    What is a smallest brick that can be put inside another cube
    touching all a faces?
    There is a elementary solution, though it seems formidable to infer a correctness.
    The resolution and explanation are even prettier in 4 dimensions.

  • Boy’s surface:
    and an
    asymmetric charcterised gif from a Harvard zoo

  • Buckyballs. The truncated icosahedron
    recently acquired new celebrity and a new name when chemists detected that
    Carbon forms molecules with a shape.

  • The
    Cheng-Pleijel point
    . Given a sealed craft bend and a tallness H,
    this indicate is a peak of a smallest aspect area cone of tallness H over
    the curve. Ben Cheng demonstrates this judgment with a assistance of a Java applet.

  • Circumcenters of triangles.
    Joe O’Rourke, Dave Watson, and William Flis
    compare formulas for computing
    the coordinates of a circle’s core from 3 range points,
    and aloft dimensional generalizations.

  • Circumnavigating
    a brick and a tetrahedron
    , Henry Bottomley.

  • Cognitive Engineering
    , Java applets for exploring tilings, symmetry, polyhedra, and
    four-dimensional polytopes.

  • Colinear points on knots.
    Greg Kuperberg
    shows that a non-trivial tangle or couple in R3
    necessarily has 4 colinear points.

  • a
    computational proceed to tilings
    . Daniel Huson investigates the
    combinatorics of periodic tilings in twin and 3 dimensions, including
    a sequence of a tilings by shapes topologically homogeneous to
    the 5 Platonic solids.

  • CSE
    . This java applet allows interactive control of a rotating
    collection of cubes.

  • Cube
    . How many smaller cubes can one order a brick into?
    From Eric Weisstein’s
    treasure trove of mathematics.

  • Cube triangulation.
    Can one order a brick into congruous and ruffle tetrahedra?
    And though a congruity assumption,
    how many aloft dimensional simplices are indispensable to triangulate a hypercube?
    For some-more on this final problem, see
    an n-dimensional cube
    , S. Finch, MathSoft,
    Asymptotically efficient
    triangulations of a d-cube
    , Orden and Santos.

  • Curvature of knots.
    Steve Fenner proves
    the Fary-Milnor
    that any smooth, simple, sealed bend in 3-space contingency have
    total span during slightest 4 pi.

  • Delta Blocks.
    Hop David discusses ideas for production building blocks done on
    the tetrahedron-octahedron space tiling decorated in Escher’s “Flatworms”.

  • Deltahedra, polyhedra with rectilinear triangle faces. From Eric Weisstein’s value trove of mathematics.

  • Densest
    packings of equal spheres in a cube
    , Hugo Pfoertner.
    With good ray-traced images of any packing.
    See also Martin
    Erren’s applet for visualizing a globe packings

  • Dodecafoam.
    A fractal stew of polyhedra fills space.

  • Dodecahedron
    , Paul Kunkel.

  • Double
    . Joel Hass investigates shapes done by soap films
    enclosing twin detached regions of space.

  • A
    dream about globe kissing numbers

  • Edge-tangent polytope illustrating Koebe’s
    theorem that any planar graph can be satisfied as a set of tangencies
    between circles on a sphere. Placing vertices during points carrying those
    circles as horizons forms a polytope with all edges tangent to a sphere.
    Rendered by POVray.

  • Escher for real and
    Escher for real
    Gershon Elber uses layered production systems to build 3d models of
    Escher’s illusions. The pretence is to make some seemingly-flat surfaces
    curve towards and divided from a viewplane.

  • All a satisfactory dice.
    Pictures of a polyhedra that can be used as dice,
    in that there is a balance holding any face to any other face.

  • Fake dissection.
    An 8×8 (64 unit) retard is cut into pieces
    which (seemingly) can be rearranged to form a 5×13 (65 unit) rectangle.
    Where did a additional section come from?
    Jim Propp asks about probable three-dimensional generalizations.
    Frederickson reserve one
    See also
    Bogomolny’s ratiocination of a 9×11 rectangle into a 10×10 square
    bamboozlement applet

  • Figure 8 tangle / horoball diagram.
    Research of A. Edmonds into a symmetries of knots,
    relating them to something that looks
    like a make-up of spheres.
    The MSRI Computing Group uses
    another horoball
    as their logo.

  • Filling space with section circles. Daniel
    Asimov asks what fragment of 3-dimensional space can be filled by a
    collection of ruffle section circles. (It might not be apparent that this
    fraction is nonzero, though a customary construction allows one to construct
    a plain torus out of circles, and one can afterwards container tori to fill space,
    leaving some unclosed gaps between a tori.) The geometry core has
    information in several places on this problem, a best being an
    describing a proceed of stuffing space by section circles

  • Five
    Platonic solids and a soccerball

  • Five
    space-filling polyhedra
    . And not a ones you’re expected meditative of,
    Guy Inchbald, reproduced from Math. Gazette 80, Nov 1996.

  • Flat
    equilateral tori.
    Can one build a polyhedral torus in that all
    faces are rectilinear triangles and all vertices have 6 incident
    edges? Probably not though this earthy indication comes close.

    flat rectilinear torus model

  • The
    flat torus in a three-sphere
    . Thomas Banchoff animates the
    Hopf fibration.

  • Flatland:
    A Romance of Many Dimensions

  • Flexible polyhedra. From Dave Rusin’s famous math pages.

  • Fractal
    . Photo by alfredo matacotta.
    See also this French page.

  • Fun with Fractals and
    the Platonic Solids
    . Gayla Chandler places models of polyhedra and
    polyhedral fractals such as a Sierpinski tetrahedron in scenic outdoor
    settings and photographs them there.

  • Gaussian
    continued fractions
    Stephen Fortescue discusses some connectors between basic
    number-theoretic algorithms and a geometry of tilings
    of 2d and 3d hyperbolic spaces.

  • Geodesic dome
    design software
    . Now we too can beget triangulations of a sphere.
    Freeware for DOS, Mac, and Unix.

  • Geometric
    by Gavin Theobald.

  • The golden
    section and Euclid’s construction of a dodecahedron
    , and
    on a dodecahedron and icosahedron
    H. Serras, Ghent.

  • Gömböc, a
    convex physique in 3d with a unaccompanied fast and a unaccompanied inconstant indicate of
    equilibrium. Placed on a prosaic surface, it always rights itself; it may
    not be a fluke that some tortoise shells are likewise shaped.
    See also Wikipedia, Metafilter, New
    York Times

  • Melinda
    Green’s geometry page
    . Green creates models of unchanging sponges
    (infinite non-convex generalizations of Platonic solids) out of plastic
    “Polydron” pieces.

  • Grid subgraphs.
    Jan Kristian Haugland looks for sets of hideaway points that induce
    graphs with high grade though no brief cycles.

  • Hebesphenomegacorona
    onna stick in space! Space Station Science pattern of
    the day. In box we don’t remember what a hebesphenomegacorona is, it’s
    one of a Johnson solids: convex polyhedra with regular-polygon faces.

  • Hecatohedra.
    John Conway discusses a probable balance groups of hundred-sided polyhedra.

  • Hedronometry.
    Don McConnell discusses equations relating a angles and face areas
    of tetrahedra. See also McConnell’s hedronometry site.

  • Helical geometry.
    Ok, renaming a hyperbolic paraboloid a “helical right triangle”
    and observant that it’s “a insubordinate substructure for new knowledge”
    seems a tiny fractious though there are some engaging cinema of shapes
    formed by compounds of these saddles.

  • Helical Gallery.
    Spirals in the
    work of M. C. Escher
    and in X-ray observations of a sun’s corona.

  • Heptomino
    Clive Tooth shows us all 108 heptominos, packaged into a 7x9x12 box.

  • Hilbert’s
    ۳rd Problem and Dehn Invariants
    How to tell possibly twin polyhedra can be dissected into any other.
    See also Walter
    Neumann’s paper
    joining these ideas with problems of
    classifying manifolds.

  • Hollow
    pyramid tetrasphere puzzle

  • Holyhedra.
    Jade Vinson solves a doubt of John Conway on a existence of
    finite polyhedra all of whose faces have holes in them
    (the Menger consume provides
    an gigantic example).

  • How many
    can one find in three-dimensional space so that all triangles
    are rectilinear or isosceles?
    One eight-point resolution is done by fixation 3 points
    on a pivot of a unchanging pentagon.
    This problem seems associated to a fact that
    any planar indicate set forms O(n7/3)
    isosceles triangles
    ; in 3 dimensions, Theta(n3) are possible
    (by generalizing a pentagon resolution above). From Stan Wagon’s
    PotW archive.

  • Human Geometry
    and Naked Geometry. The
    human form as a building retard of incomparable geometric figures, by Mike

  • Ideal
    hyperbolic polyhedra

    ray-traced by Matthias Weber.

  • Guy Inchbald’s
    polyhedra pages
    Stellations, hendecahedra, duality, space-fillers, quasicrystals, and more.

  • IFS and L-systems.
    Vittoria Rezzonico grows fractal broccoli and Sierpinski pyramids.

  • Interactive
    fractal polyhedra
    , Evgeny Demidov.

  • The International
    Bone-Roller’s Guild
    ponders the
    polyhedra that can act as satisfactory dice, since all faces are symmetric to
    each other.

  • Intersecting brick diagonals.
    Mark McConnell asks for a explanation that, if a convex polyhedron
    combinatorially homogeneous to a brick has 3 of a four
    body diagonals assembly during a point, afterwards a fourth one meets
    there as well. There is apparently some tie to toric varieties.

  • Java lamp, S. M.

  • Johnson Solids, convex polyhedra with unchanging faces. From Eric Weisstein’s
    treasure trove of mathematics.

  • Sándor Kabai’s
    mathematical graphics
    , essentially polyhedra and 3d fractals.

  • Aaron Kellner Linear Sculpture.
    Art in a form of geometric tangles of steel and timber rods.

  • Kelvin surmise counterexample.
    Evelyn Sander forwards news about a find by Phelan and Weaire
    of a improved proceed to assign space
    into equal-volume low-surface-area cells.
    Kelvin had conjectured that a truncated octahedron supposing a optimal
    solution, though this incited out not to be true.
    See also Ruggero Gabbrielli’s comparison of equal-volume partitions and
    foam models

  • Kepler-Poinsot
    , concave polyhedra with star-shaped faces. From Eric
    Weisstein’s value trove of mathematics. See also
    H. Serras’
    page on Kepler-Poinsot solids

  • Knot pictures. Energy-minimized well-spoken and polygonal knots, from the
    knot evolver, Y. Wu, U. Iowa.

  • Mathematical imagery by Jos Leys.
    Knots, Escher tilings, spirals, fractals, round inversions, hyperbolic
    tilings, Penrose tilings, and more.

  • Louis Bel’s povray galleries:
    polyhèdres réguliers
    more knots.

  • Maille Weaves.
    Different repeated patterns done by associated circles along a craft in space,
    as used for creation sequence mail. Along with some linear patterns for
    jewelry chains.

  • Maple
    polyhedron gallery

  • Martin’s pretty
    . Simulation of particles repulsion any other on the
    sphere produces good triangulations of a surface.

  • Mathematica Graphics Gallery: Polyhedra

  • Mathenautics. Visualization of 3-manifold geometry during a Univ. of Illinois.

  • MatHSoliD
    Java animation of planar unfoldings of a Platonic and Archimedean polyhedra.

  • Minesweeper
    on Archimedean polyhedra
    , Robert Webb.

  • Minimax effervescent tortuous appetite of globe eversions.
    Rob Kusner, U. Mass. Amherst.

  • Minimizing
    surface area to volume ratio in a cube

  • Maximum volume
    arrangements of points on a sphere
    , Hugo Pfoertner.

  • Miquel’s six
    circles in 3d
    Reinterpreting a matter about intersecting circles to be about
    inscribed cuboids.

    Miquel'sSix-Circle Theorem

  • Modeling
    mollusc shells with logarithmic spirals
    , O. Hammer, Norsk
    Net. Tech. Also includes a list of logarithmic turn links.

  • Models of Platonic solids
    and associated symmetric polyhedra.

  • Nested
    Klein bottles
    . From a London Science Museum gallery, by proceed of Boing
    . Topological glassware by Alan Bennett.

  • Netlib polyhedra.
    Coordinates for unchanging and Archimedean polyhedra,
    prisms, anti-prisms, and more.

  • Nine.
    Drew Olbrich discovers a associahedron by uniformly spacing 9 points
    on a globe and dualizing.

  • No cubed cube.
    David Moews offers a lovable explanation that no brick can be divided into smaller
    cubes, all different.

  • T. Nordstrand’s
    gallery of surfaces

  • Not. AMS
    Cover, Apr. 1995
    . This painting for an letter on geometric
    tomography depicts objects (a cuboctahedron and mangled rhombic
    dodecahedron) that costume themselves as unchanging tetrahedra
    by carrying a same breadth duty or cat-scan image.

  • Objects that can't be taken detached with twin hands.
    J. Snoeyink, U. British Columbia.

  • Occult correspondences of a Platonic solids.
    Some pointless thoughts from

  • Orthogonal dissimilar knots.
    Hew Wolff asks questions about a smallest sum length, or a smallest volume of a rectilinear box, indispensable to form opposite knots as three-dimensional polygons regulating usually integer-length axis-parallel edges.

  • Packing
    circles in circles
    and circles on a sphere,
    Jim Buddenhagen.
    Mostly about optimal make-up though includes also some nonoptimal spiral
    and pinwheel packings.

  • Packing
    Tetrahedrons, and Closing in on a Perfect Fit
    . Elizabeth Chen and
    others use experiments on hundreds of DD bones to pound previous
    records for make-up density.

  • Pairwise
    touching hypercubes
    . Erich Friedman asks how to assign a section cubes
    of an a*b*c-unit rectilinear box into as many connected polycubes as
    possible with a common face between any span of polycubes.
    He lists both ubiquitous top and reduce end as functions of a, b, and
    c, and specific constructions for specific sizes of box.
    I’ve seen a same doubt asked for d-dimensional hypercubes
    formed out of 2^d section hypercubes; there is a reduce firm of roughly
    ۲d/2 (from embedding a 2*2d/2*2d/2 box
    into a hypercube)
    and an top firm of O(2d/2 sqrt d)
    (from computing how many cubes contingency be in a polycube
    to give it adequate faces to hold all a others).

  • Pappus
    on a Archimedean solids
    . Translation of an mention of a fourth century
    geometry text.

  • Penumbral shadows of polygons
    form projections of four-dimensional polytopes.
    From a Graphics Center’s graphics archives.

  • Pictures of 3d and 4d unchanging solids, R. Koch, U. Oregon.
    Koch also provides some
    ۴D unchanging plain cognisance applets.

  • The
    Platonic solids
    . With Java viewers for interactive manipulation. Peter Alfeld, Utah.

  • Platonic
    solids and Euler’s formula
    . Vishal Lama shows how a regulation can be
    used to uncover that a informed 5 Platonic solids are a usually ones

  • Platonic
    remade by Michael Hansmeyer regulating subdivision-surface
    algorithms into shapes
    resembling radiolarans.
    See also Boing Boing discussion.

  • Platonic Universe,
    Stephan Werbeck. What shapes can we form by gluing unchanging dodecahedra

  • Polygons as projections of polytopes.
    Andrew Kepert answers a doubt of
    George Baloglou on possibly any planar figure done by a convex
    polygon and all a diagonals can be done by raised a
    three-dimensional convex polyhedron.

  • Polyhedra.
    Bruce Fast is building a library of images of polyhedra.
    He describes some of a unchanging and semi-regular polyhedra,
    and lists names of many some-more including a Johnson solids
    (all convex polyhedra with unchanging faces).

  • Polyhedra Blender.
    Mathematica program and Java-based interactive web gallery for what demeanour like
    Minkowski sums of polyhedra. If a inputs to a Minkowski
    sums were line segments, cubes, or zonohedra, a regulation would be again
    zonohedra, though a ability to supply other inputs allows some-more general
    polyhedra to be formed.

  • Polyhedra
    , V. Bulatov.

  • Polyhedra
    Many regular-polyhedron compounds, rendered in povray by Alexandre Buchmann.

  • A
    polyhedral analysis
    . Ken Gourlay looks during a Platonic solids and
    their stellations.

  • Polyhedron,
    polyhedra, polytopes, … – Numericana

  • Polyhedron challenge: cuboctahedron.

  • Polyominoids,
    connected sets of squares in a 3d cubical lattice.
    Includes a Java applet as good as non-animated description.
    By Jorge L. Mireles Jasso.

  • The
    Pretzel Page
    . Eric Sedgwick uses charcterised cinema of rambling pretzel knots
    to daydream a postulate about Heegard splittings
    (ways of dividing a formidable topological space into twin elementary pieces).

  • Prince
    Rupert’s Cube
    . It’s probable to pull a incomparable brick by a hole
    drilled into a smaller cube. How most larger? 1.06065… From Eric
    Weisstein’s value trove of mathematics.

  • Prince
    Rupert’s tetrahedra?
    One tetrahedron can be wholly contained in
    another, and nonetheless have a incomparable sum of corner lengths. But how most larger?
    From Stan Wagon’s
    PotW archive.

  • Programming for 3d
    , T. Longtin. Tensegrity structures, disfigured torus space frames,
    Moebius wire rigging assemblies, jigsaw nonplus polyhedra, Hilbert fractal helices,
    herds of turtles, and more.

  • Proofs of Euler’s Formula.
    V-E+F=2, where V, E, and F are respectively a numbers of
    vertices, edges, and faces of a convex polyhedron.

  • Pseudospherical surfaces.
    These surfaces are equally “saddle-shaped” during any point.

  • Quaquaversal
    Tilings and Rotations
    . John Conway and Charles Radin report a
    three-dimensional generalization of a pinwheel tiling, a mathematics
    of that is messier due to a noncommutativity of three-dimensional

  • Quark constructions.
    The sun4v.qc Team investigates polyhedra that fit together
    to form a modular set of building blocks.

  • Quark
    . An fleeting outward arrangement of geometric art, in Princeton,
    New Jersey. From Ivars Peterson’s MathTrek.

    and aperiodic tilings
    , A. Zerhusen, U. Kentucky.
    Includes a good outline of how to make 3d aperiodic tiles
    from zometool pieces.

  • Qubits, modular geometric building
    blocks by designer Mark Burginger, desirous by Fuller’s geodesic domes.

  • Ram’s Horn
    cardboard indication of an engaging 3d turn figure restrained by a helicoid
    and twin nested cones.

  • Regard
    mathématique sur Bruxelles
    . Student plan to photograph
    city facilities of mathematical seductiveness and indication them in Cabri.

  • Regular
    polyhedra as intersecting cylinders
    Jim Buddenhagen exhibits ray-traces of a shapes done by
    extending half-infinite cylinders around rays from a center
    to any zenith of a unchanging polyhedron.
    The range faces of a ensuing unions form
    combinatorially homogeneous complexes to those of a twin polyhedra.

  • Regular solids.
    Information on Schlafli symbols, coordinates, and duals
    of a 5 Platonic solids.
    (This page’s pretension says also Archimedean solids, though we don’t see many of
    them here.)

  • Rhombic
    , concave rhombus-faced polyhedra that tile space,
    R. Towle.

  • Riemann Surfaces and a Geometrization of 3-Manifolds,
    C. McMullen, Bull. AMS 27 (1992).
    This expository (but really technical) letter outlines Thurston’s
    technique for anticipating geometric structures in 3-dimensional topology.

  • Rob’s
    polyhedron models
    , done with a assistance of his program

  • Robinson Friedenthal polyhedral explorations.
    Geometric sculpture.

  • Rolling
    . Dave Boll investigates Hamiltonian paths on (duals of)
    regular polyhedra.

  • Rudin’s
    example of an unshellable triangulation
    In this resolution of a large tetrahedron into tiny tetrahedra,
    every tiny tetrahedron has a zenith interior to a face of a big
    tetrahedron, so we can’t mislay any of them though mixing a hole.
    Peter Alfeld, Utah.

  • Ruler and Compass.
    Mathematical web site including special sections on the
    geometry of
    of polytopes

  • The
    Schläfli Double Six
    A poetic photo-essay of models of this configuration,
    in that twelve lines any accommodate 5 of thirty points.
    Unfortunately usually a initial page seems to be archived…
    (This site also referred to
    related configurations involving 27 lines assembly possibly 45 or
    ۱۳۵ points, though didn’t report any mathematical details.
    For serve descriptions of all of these, see Hilbert and
    Cohn-Vossen’s “Geometry and a Imagination”.)

  • In hunt of a ideal knot.
    Piotr Pieranski relates an iterative timorous heuristic to find the
    minimum length unit-diameter wire that can be used to tie a given knot.

  • Seashell spirals.
    Xah Lee examines a shapes of several genuine seashells, and offers prize
    money for formulas duplicating them.

  • The Sierpinski Tetrahedron, everyone’s
    favorite 3 dimensional fractal.
    Or is it a fractal?

  • SingSurf
    software for calculating unaccompanied algebraic curves and surfaces, R. Morris.

  • Six-regular toroid.
    Mike Paterson asks possibly it is probable to make a torus-shaped polyhedron
    in that accurately 6 rectilinear triangles accommodate during any vertex.

  • Skewered lines.
    Jim Buddenhagen records that 4 lines in ubiquitous position in R3
    have accurately twin lines channel them all, and asks how this generalizes
    to aloft dimensions.

  • Soap films on knots. Ken Brakke, Susquehanna.

  • Soddy’s Hexlet,
    six spheres in a ring tangent to 3 others,
    and Soddy’s
    Bowl of Integers
    , a globe make-up mixing forever many hexlets,
    from Mathworld.

  • Solution
    of Conway-Radin-Sadun problem
    Dissections of combinations of unchanging dodecahedra, unchanging icosahedra,
    and associated polyhedra into rhombs that tile space. By Dehn’s resolution to
    Hilbert’s third problem this is unfit for particular dodecahedra
    and icosahedra, though Conway,
    Radin, and Sadun
    showed that certain combinations could work.
    Now Izidor Hafner shows how.

  • Solution
    to problem 10769
    . Apparently problems of coloring a points of a
    sphere so that quadratic points have opposite colors (or so that each
    set of coordinate basement vectors has mixed colors) has some relevance
    to quantum mechanics; see also papers
    quant-ph/9905080 and
    (on coloring only a receptive points on a sphere), as good as this
    four-dimensional construction
    of an peculiar series of basement sets in that any matrix appears an even
    number of times, display that one can’t tone a points on a
    four-sphere so that any basement set has accurately one black point.

  • Soma cube

  • The soma brick page and pentomino page, J. Jenicek.

  • Some images done by Konrad Polthier.

  • Some cinema of symmetric tensegrities.

  • SpaceBric building blocks
    and Windows program done on a tiling of 3d space by congruent

  • Sphere make-up and kissing numbers.
    How should one arrange circles or spheres
    so that they fill space as densely as possible?
    What is a limit series of spheres that can simultanously touch
    another sphere?

  • Spherical
    Julia set with dodecahedral symmetry
    discovered by McMullen and Doyle in their work on
    quintic equations and rendered by
    Don Mitchell.
    Update 12/14/00: I’ve mislaid a large chronicle of this pattern and can’t find
    DonM anywhere on a net — can anyone help?
    In a meantime, here’s a couple to

  • The sphericon,
    a convex figure with one winding face and twin semicircular edges that can
    roll with a wobbling suit in a true line.
    See also
    national bend bank sphericon page
    the MathWorld
    sphericon page
    the Wikipedia sphericon page,
    Differential Geometry of a Sphericon
    , and
    building a

  • Spiral
    . Photo of a building in Iraq, partial of a web letter on the
    geometry of cyberspace.

  • Spiraling
    Sphere Models
    . Bo Atkinson studies a geometry of a plain of
    revolution of an Archimedean spiral.

  • Spring
    into action
    . Dynamic origami. Ben Trumbore, done on a indication by Jeff
    Beynon from Tomoko Fuse’s book Spirals.

  • Square Knots. This letter by Brian Hayes for American Scientist
    examines how expected it is that a random
    lattice polygon is knotted.

  • Stella and Stella4d,
    Windows program for visualizing unchanging and semi-regular polyhedra and
    their stellations in 3 and 4 dimensions, morphing them into any other, sketch unfolded nets for
    making paper models, and exporting polyhedra to several 3d pattern packages.

  • Sterescopic polyhedra
    rendered with POVray by Mark Newbold.

  • Steve’s sprinklers.
    An engaging 3d polygon done of copper siren forms several symmetric 2d shapes
    when noticed from opposite directions.

  • Subdivision
    . Strange diatom-like shapes done by varying the
    parameters of a spline aspect filigree excellence intrigue outward their
    normal ranges.

  • Symmetries of torus-shaped polyhedra

  • The Szilassi Polyhedron.
    This polyhedral torus, detected by
    , has 7 hexagonal faces, all adjacent to any other.
    It has an pivot of 180-degree symmetry; 3 pairs of faces are congruent
    leaving one unpaired hexagon that is itself symmetric.
    Ace has some-more images
    as good as a downloadable unfolded pattern
    for creation your possess copy.
    See also Dave Rusin’s page on
    tori with few vertices
    Peterson’s MathTrek article

  • Tales of the
    , from Pythagoras to Plato to Poincaré. John
    Baez, Reese Prosser Memorial Lecture, Dartmouth, 2006.

  • Tangencies
    of circles and spheres
    . E. F. Dearing provides formulae for the
    radii of Apollonian circles, and equivalent three-dimensional problems.

  • Tensegrity zoology.
    A catalog of fast structures done out of springs,
    somehow mixing a quantum speculation of what used to be described as time.

  • Tetrahedra
    . Mathematica doing of a Chen-Engel-Glotzer packing
    of space by unchanging tetrahedra, a densest famous such make-up to date.

  • Tetrahedrons and spheres.
    Given an capricious tetrahedron, is there a globe tangent to any of a edges?
    Jerzy Bednarczuk, Warsaw U.

  • Tetrahedra classified
    by their bad angles
    From “Dihedral end for mesh
    generation in high dimensions

  • These twin cinema by Richard Phillips
    are from a now-defunct maths with photographs website.
    The funnel is (Phillips thinks) somewhere in North Nottinghamshire, England.
    A identical collection of Phillips’
    mathematical photos is now accessible on CD-ROM.


  • Thoughts on a series six.
    John Baez contemplates a symmetries of a icosahedron.

  • Three cubes to one.
    Calydon asks possibly 9 pieces is optimal for this ratiocination problem.

  • ۳D-Geometrie.
    T. E. Dorozinski provides a gallery of images of 3d polyhedra,
    ۲d and 3d tilings, and subdivisions of winding surfaces.

  • ۳d-XplorMath
    Macintosh program for visualizing curves, surfaces, polyhedra,
    conformal maps, and other planar and three-dimensional mathematical objects.

  • Three-dimensional models done on a works of M. C. Escher

  • The
    three dimensional polyominoes of minimal area
    , L. Alonso and
    R. Cert, Elect. J. Combinatorics.

  • Three dimensional turtle speak outline of a dodecahedron. The dodecahedron’s outline is “M40T72R5M40X63.435T288X296.565R5M40T72M40X63.435T288X296.565R4”; isn’t that helpful?

  • ۳D bizarre attractors and identical objects, Tim Stilson, Stanford.

  • Three untetrahedralizable objects

  • Tilable
    Patrick Snels creates two-dimensional images that tile a craft to
    form 3d-looking views including some engaging Escher-like warped
    See also his even some-more Escherian tesselations page.

  • Tiling with 4 cubes.
    Torsten Sillke summarizes regulation and conjectures on
    the problem of tiling 3-dimensional boxes with a tile
    formed by gluing 3 cubes onto 3 adjacent faces
    of a fourth cube.

  • Tiling with
    notched cubes
    . Robert Hochberg and Michael Reid vaunt an unboxable
    reptile: a polycube that can tile a incomparable duplicate of itself, though can’t
    tile any rectilinear block.

  • Toroidal tile for tessellating three-space, C. Séquin, UC Berkeley.

  • Triangulating 3-dimensional polygons.
    This is always probable (with exponentially many Steiner points)
    if a polygon is unknotted, though NP-complete if no Steiner points are allowed.
    The explanation uses gadgets in that quadrilaterals are
    stacked like Pringles to form wires.

  • Triangulation numbers. These systematise a geometric structure of
    viruses. Many viruses are done as simplicial polyhedra consisting of 12
    symmetrically placed grade 5 vertices and some-more grade 6 vertices;
    the series represents a stretch between grade 5 vertices.

  • Triply
    orthogonal surfaces
    , Matthias Weber.

  • Truncated
    . Hop David has a good pattern of Coxeter’s unchanging sponge
    {۶,۴|۴}, done by withdrawal out a retard faces from a tiling of space by truncated octahedra.

  • Truncated
    Trickery: Truncatering
    Some truncation family among a Platonic solids and their friends.

  • Tune’s polyhedron models.
    Sierpinski octahedra, stellated icosahedra, interlocking
    zonohedron-dissection puzzles, and more.

  • Turkey
    . A brick ratiocination nonplus from IBM research.

  • Tuvel’s
    Polyhedra Page
    Tuvel’s Hyperdimensional
    Information and images on concept polyhedra and aloft dimensional polytopes.

  • ۲۷ lines on a Clebsch cubic, Matthias Weber.

  • The uniform net
    . An engaging clear structure done by make-up square
    and octagonal helices.

  • Uniform polyhedra.
    Computed by Roman Maeder regulating a Mathematica
    implementation of a process of Zvi Har’El.
    Maeder also includes alone a pattern of the
    ۲۰ convex uniform polyhedra, and descriptions of the
    stellations of a icosahedra

  • Uniform
    polyhedra in POV-ray format
    , by Russell Towle.

  • Uniform
    , R. Morris. Rotatable 3d java perspective of these polyhedra.

  • An uninscribable 4-regular polyhedron.
    This figure can not be drawn with all a vertices on a unaccompanied sphere.

  • Visualization of a Carrillo-Lipman Polytope. Geometry outset from a coexisting comparison of mixed DNA or protein sequences.

  • Volume of a torus.
    Paul Kunkel describes a elementary and discerning proceed of anticipating a regulation for a
    torus’s volume by relating it to a cylinder.

  • Volumes in
    . Volumes of several unchanging and semi-regular polyhedra,
    scaled according to stamped tetrahedra.

  • Volumes of pieces of a dodecahedron.
    David Epstein (not me!) wonders because together slices by a layers
    of vertices of a dodecahedron furnish equal-volume chunks.

  • vZome
    zometool pattern program for OS X and Windows.
    (Warning, web site might be down on off-hours.)

  • The
    Water Cube
    swimming venue during a 2008 Beijing Olympics uses the
    Weaire-Phelan froth (a assign of 3d space into equal-volume cells with
    the smallest famous aspect area per section volume) as a basement of a structure.

  • Waterman polyhedra,
    formed from a convex hulls of centers of points nearby a start in an
    alternating lattice.
    See also Paul
    Bourke’s Waterman Polyhedron page

  • Matthias
    Weber’s gallery of ray-traced mathematical objects
    , such as minimal
    surfaces floating in ponds.

  • Why doesn’t Pick’s postulate generalize?
    One can discriminate a volume of a two-dimensional polygon with integer
    coordinates by counting a series of integer points in it and on its
    boundary, though this doesn’t work in aloft dimensions.

  • Why “snub cube”?
    John Conway provides a doctrine on polyhedron nomenclature and etymology.
    From a geometry.research archives.

  • Zometool. The 31-zone constructional complement for constructing
    “mathematical models, from tilings to hyperspace projections, as good as
    molecular models of quasicrystals and fullerenes, and architectural
    space support structures”.

  • Zonohedra and zonotopes. These centrally
    symmetric polyhedra yield another proceed of bargain the
    combinatorics of line arrangements.

  • A zoo of surfaces.

  • Frank Zubek’s
    Elusive Cube
    . Magnetic tetrahedra bond to form dissections of
    cubes and many other shapes.

From a Geometry Junkyard,
and recreational geometry pointers.
Send email if you
know of an suitable page not listed here.
David Eppstein,
Theory Group,
UC Irvine.
from a common source file.

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