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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

aperiodically.

Ludwig Danzer speaks during NYU on

various aperiodic 3d tilings including Conway’s

biprism. - 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

and

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

conjecture, 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.

Matthew

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:

Wikipedia,

MathWorld,

Geometry

Center,

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

Lab, 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 R^{3}

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

logo. This java applet allows interactive control of a rotating

collection of cubes. - Cube

Dissection. 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

Triangulating

an n-dimensional cube, S. Finch, MathSoft,

and

Asymptotically efficient

triangulations of a d-cube, Orden and Santos. - Curvature of knots.

Steve Fenner proves

the Fary-Milnor

theorem 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

measures, Paul Kunkel. - Double

bubbles. 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

beyond

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.

Greg

Frederickson reserve one.

See also

Alexander

Bogomolny’s ratiocination of a 9×11 rectangle into a 10×10 square and

Fibonacci

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

diagram 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

article

describing a proceed of stuffing space by section circles (discontinuously). - Five

Platonic solids and a soccerball. - Five

space-filling polyhedra. And not a ones you’re expected meditative of,

either.

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. - 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

broccoli. 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

Dissections by Gavin Theobald. - The golden

section and Euclid’s construction of a dodecahedron, and

more

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

Packings.

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

points 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(n^{7/3})

isosceles triangles; in 3 dimensions, Theta(n^{3}) 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

Naylor. - 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

isohedra:

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.

Christensen. - 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

JavaView

foam models. - Kepler-Poinsot

Solids, 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

ming

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:

les

polyhèdres réguliers,

knots,

and

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

polyhedra. 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. - 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

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

Anders

Sandberg. - 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

۲^{d/2}(from embedding a 2*2^{d/2}*2^{d/2}box

into a hypercube)

and an top firm of O(2^{d/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

possible. - Platonic

solids 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

face-to-face? - 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

collection, V. Bulatov. - Polyhedra

exhibition.

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

modeling, 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

rotations. - Quark constructions.

The sun4v.qc Team investigates polyhedra that fit together

to form a modular set of building blocks. - Quark

Park. An fleeting outward arrangement of geometric art, in Princeton,

New Jersey. From Ivars Peterson’s MathTrek.Quasicrystals

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

spirallohedra, 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

Stella. - Robinson Friedenthal polyhedral explorations.

Geometric sculpture. - Rolling

polyhedra. 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

polyhedrons and

geometry

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 R^{3}

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

quant-ph/9911040

(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

applet. - 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

tetrahedra. - 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

McMullen’s

rendering. - 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

the

national bend bank sphericon page,

the MathWorld

sphericon page,

the Wikipedia sphericon page,

The

Differential Geometry of a Sphericon, and

building a

sphericon. - Spiral

tower. 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

kaleidoscope. 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

L.

Szilassi, 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.

Tom

Ace has some-more images as good as a downloadable unfolded pattern

for creation your possess copy.

See also Dave Rusin’s page on

polyhedral

tori with few vertices and

Ivars’

Peterson’s MathTrek article. - Tales of the

dodecahedron, 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

packing. 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

perspectives.

Patrick Snels creates two-dimensional images that tile a craft to

form 3d-looking views including some engaging Escher-like warped

perspectives.

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

Octahedra. 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

stuffing. A brick ratiocination nonplus from IBM research. - Tuvel’s

Polyhedra Page and

Tuvel’s Hyperdimensional

Page.

Information and images on concept polyhedra and aloft dimensional polytopes. - ۲۷ lines on a Clebsch cubic, Matthias Weber.
- The uniform net

(۱۰,۳)-a. 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

۵۹

stellations of a icosahedra. - Uniform

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

polyhedra, 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

synergetics. 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,

computational

and recreational geometry pointers.

Send email if you

know of an suitable page not listed here.

David Eppstein,

Theory Group,

ICS,

UC Irvine.

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filtered

from a common source file.

Article source: http://www.ics.uci.edu/~eppstein/junkyard/3d.html

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