If you want to learn a lot about regular polytopes, read this book by. The dodecahedron occurs as cells in the 120 cell and the runcinated 120 cell a dodecahedron can be augmented by a pentagonal pyramid to produce an augmented dodecahedron j58, one of the johnson solids. A 120cell is sitting in x, y, z, wspace with one of its dodecahedral faces in the 3plane. Klein discovered that the rotation groups of the tetrahedron, cube and its dual octahedron, and icosahedron and its dual dodecahedron are none other than the alternating and symmetric groups a4, s4, and a5 respectively. This object is the truncated 120 cell, a 3d projection or shadow of a 4dimensional shape. Thats how you go from the tessellation of the 120 cell. The bases of the triangular prisms come together in groups of four, making regular tetrahedra in the places corresponding to the 120 cell s vertices. Jun 03, 2010 this short computer graphics animation presents the regular 120 cell. My best interpretation of descriptions ive read and heard about is that this is essentially the 3d shadow of 120.
Regular icosahedron wikimili, the best wikipedia reader. In perspective projection, viewed on top of a pentagonal face, the regular dodecahedron can be seen as a linearedged schlegel diagram, or stereographic projection as a spherical polyhedron. A reverse phone number search can be very helpful in many situations. A rectified dodecahedron forms an icosidodecahedron. It encodes logical qubits and n144 physical qubits and. It is one of the five platonic solids, and the one with the most sides. Apr 19, 2016 i might make a more technical second post on this topic explaining in detail the action, but suffice it to say that we have an action by a group that has 120 elements, so that if we quotient out this 120 cell by the action, we end up with just one dodecahedron with the faces identified the way we want them to opposite faces identified by a twist. Feb 07, 2015 platonic solids which has more volume, a dodecahedron or an icosahedron, both having the same edge length. Pentakisdodecahedron dictionary definition pentakis. Thats how you go from the tessellation of the 120cell to the poincare homology sphere dodecahedral space. A regular dodecahedron or pentagonal dodecahedron is a dodecahedron that is regular, which is composed of twelve regular pentagonal faces, three meeting at each vertex. The most familiar dodecahedron is the regular dodecahedron, which is a platonic solid. Cut out each panel along the circle not the pentagon 3.
In this projection, not all of the 120 dodecahedra of the 120cell are visible. Feb 17, 2017 dodecahedron, a science fiction story set in the near future. The hyper dodecahedron is composed of 120 dodecahedra, 720 pentagons, 600 vertices and 1200 edges. Every vertex of this expanded 120cell is identicalthe meeting place of one dodecahedron. The 120cell spherical dodecahedron tile the surface of a hypersphere. Zome tool 120cell hyperdodecahedron knowe learning. Check out the 4dimensional analogue of the dedocahedron. The 120 cell is named for its 120 dodecahedral faces. The 120cell is named for its 120 dodecahedral faces. Which has more volume, a dodecahedron or icosahedron. Europe pmc is an archive of life sciences journal literature.
At the end of this past year in my geometry class we decided to tackle the goal of building a 120 cell hyperdodecahedron. Dodecahedron model humble independent school district. The 600 tetrahedra correspond with the 600 tetrahedra in the 600cell, although here they are in dual orientation. The boundary of the 120 cell is composed of 120 dodecahedral cells with 4 meeting at each vertex. The 120cell can be constructed by joining 12 dodecahedra to a starting dodecahedron and rotating each new dodecahedron 36 degrees along each common face. It has 12 faces, 20 vertices, 30 edges, and 160 diagonals 60 face diagonals, 100 space diag. These projections are also used in showing the fourdimensional 120 cell, a regular 4dimensional polytope, constructed from 120. Authors answer 120 dodecahedron i read encyclopedias. This is the result of taking the edges of the polytope, radially projecting them onto the unit 3sphere in 4dimensional space, then stereographically projecting the result. It has 12 faces, 20 vertices, 30 edges, and 160 diagonals 60 face diagonals, 100 space diagonals. It is also known as the hyperdodecahedron or hecatonicosachoron, and is composed of 120 dodecahedra, with 3 to an edge, and 720 pentagons coxeter 1973, p. The rhombic dodecahedron can be seen as a degenerate pyritohedron where the 6 special edges have been reduced to zero length, reducing the pentagons into rhombic faces.
If you wanted to represent a number 5 as a 2d shape youd make a pentagon, as a 3d shape youd make dodecahedron, as a 4d shape youd have a hyperdodecahedron. Heavily frontloaded, the first couple sections are completely perfect but by the end the story seems more or less to be running out its prescribed motions with diminishing interest and complexity until it dissipates entirely into the emptiness of the uncaring universe. We will focus on her analysis of one of the hypersolids, the 120 cell, with boundary consisting of 120 dodecahedra. Dodecahedral construction of the poincare homology sphere. S t ep 1 s t ep 2 s t ep 3 s t ep 4 s t ep 5 s t ep 6 fold tips.
The hyperdodecahedron mathematicians call it the 120cell with 120. The pentagons are reduced in size and new hexagonal faces are added in place of all the original edges. The wonderful world of mathematics constructing the 120 cell. Sep 17, 2015 repeat this with all immediately neighboring dodecahedra to get five more intertwined annuli of dodecahedra. Many of these viruses exhibit exquisitely symmetric organization. The spherical dodecahedron fit together snugly because their edge angles are exactly 120. Its a twelvesided polygon, and if you play role playing games, such as dungeons and dragons, you most likely have used one. In his famous lectures on the icosahedron, klein made a meal of. Let p be the rightangled dodecahedron or 120 cell in hyperbolic space, and let w be the group generated by reflections across codimensionone faces of p. In total there are 20 corners also called vertices and 30 edges altogether. My favorite numbers university of california, riverside. In 4space all of the dodecahedra have correct geometry, however as a consequence of the projection into 3space all but the central dodecahedra are distorted. A pyritohedron is a dodecahedron with pyritohedral t h symmetry.
Mobile of the fourth dimension the mathematical tourist. A notable feature of this family is that the ratio between the volumes of the 4manifolds and their boundary components is constant and, in particular, bounded. We prove that if gamma is a torsionfree subgroup of minimal index in w, then the corresponding hyperbolic manifold hngamma is determined up to homeomorphism by gamma modulo symmetries of. The 120cell is to the dodecahedron as a tesseract mathworld. Small covers of the dodecahedron and the 120cell nasaads. Henrysegerman saulschleimer oklahomastateuniversity. Attach a piece of monofilament fishing line to the top of the globe in order to hang it from the ceiling.
Allison morris is a postgrad math student who has just discovered how to travel through time. I first came across this book in a wh smith sale in 1981 i was working there at the time my first job after leaving school and was immediately intrigued by it. This polyhedron can be formed from a dodecahedron by truncating cutting off the corners so the pentagon faces become decagons and the corners become triangles it is used in the cell transitive hyperbolic spacefilling tessellation, the bitruncated icosahedral honeycomb area and volume. Alicia boole stott and the 120cell stony brook mathematics.
May 28, 2018 one known example is the orientable closed davis manifold obtained from identifying opposing dodecahedra in the 120 cell, viewed as a 4polytope. This picture illustrates how the layers stack up when the polytope is placed with one solid face in x, y, z space. Irrespective of their shape and size, the underlying theme in all these viruses is that the virus structure is designed to contain and protect the viral genome, and deliver it to a specific host cell for subsequent replication of the virus. The boundary of the 120cell is composed of 120 dodecahedral cells with 4 meeting at each vertex.
Schlegel diagram vertices and edges type convex regular 4 polytope schlafli symbol 5,3,3. The tksst gift guide this model of a hyperdodecahedron. Its named for its 120 dodecahedral faces, which appear skewed in this projection, but of course in 4space they are all perfect. This short computer graphics animation presents the regular 120cell. The 120 cell is a 4d structure made of 120 regular dodecahedra. The small stellated dodecahedron is also a faceting of the icosahedron, though it is more commonly thought of as a stellation of the dodecahedron. The spherical dodecahedron is just a dodecahedral piece of a hypersphere. Regular dodecahedron wikimili, the best wikipedia reader. It is used in the cell transitive hyperbolic spacefilling tessellation, the bitruncated icosahedral honeycomb. A tiling derived from a projection of the small rhombicuboctahedron. This model of a hyperdodecahedron or 120cell makes for a stunning puzzle.
The rhombic dodecahedron packs together to fill space. As a little boy i found a book at a funky 60s craft store. It is constructed as a chamfer geometry edgetruncation of a regular dodecahedron. Fold along each line of the pentagon within the circle. Alicia is said to have learned the first two books of euclid, and no more. When folding the edges in step 3, use a straight edge, such as a credit card as your guide. The dodecahedron has 5 12 60 rotational symmetries, since a rotation can carry a face to any of the 12 faces in 5 di erent ways. Small covers of the dodecahedron and 120 cell in this section, w e describe an algorithm for enumerating equiv alence classes of small cov ers for general p, and apply it to the dodecahedron and. The truncated 120 cell or truncated hecatonicosachoron is a uniform 4polytope, constructed by a uniform truncation of the regular 120 cell 4polytope. Half the department gathers in secret to test her theories to some measure of success but in the process damages the very fabric of spacetime. The pattern of these 20 vertex coordinate numbers is rather interesting when written in terms of the golden mean 15 2.
All six annuli together form one half of the 120 cell, the rest just being the complement in the 3sphere of what we already have. A mathematical coloring book by marshall hampton dedicated to violet hampton version 0. Clive tooth makes polyhedra out of his deep and inscrutable singular name. The tksst gift guide this model of a hyperdodecahedron or. We explore a distance3 homological css quantum code, namely the small stellated dodecahedron code, for dense storage of quantum information and we compare its performance with the distance3. Picture of the book cover it can be drawn or printed from the computer. It can be thought of as the 4dimensional analog of the dodecahedron. Feb 8, 2014 check out the 4dimensional analogue of the dedocahedron, the hyper dodecahedron or the 120 cell. The regular dodecahedron is a platonic solid bounded by 12 regular pentagons.
This polyhedron can be formed from a dodecahedron by truncating cutting off the corners so the pentagon faces become decagons and the corners become triangles. The article whataretheconditionsforapolygontobetessellated helped me understand which polyhedron are and. This steelbronze representation of the 120 cell was created by bathsheba grossman. The same question can be asked of the cube and octahedron, and the following discussion applies just as well to them. I tried reproducing it with 12 sided dices, or in blender, but it looked impossible, because there is a growing gap inbetween faces.
It was a detailed guide on how to make all sorts of geometric solids. Recalling that in this shape we have four cells per vertex and three per edge, a little calculation yields the following table. This shadow of it has the form of one large dodecahedron filled in with 119 smaller dodecahedra. We prove that if gamma is a torsionfree subgroup of minimal index in w, then the corresponding hyperbolic manifold hngamma is determined up to homeomorphism by gamma modulo symmetries of p. Here is an image of just the vertices and edges of the 120 cell. The regular dodecahedron shares its 20 vertices with that of 20 vertices of the 120 polyhedron type iii. Now add one dodecahedron on each face of the first dodecahedron. There are also three regular star dodecahedra, which are constructed as stellations of the convex form. Its named for its 120 dodecahedral faces, which appear skewed in this projection, but of course in 4space they are all perfect regular dodecahedra. This piece is a three dimensional representation of the four dimensional analog of the dodecahedron. Its tempting to think that the icosahedron is bigger, because it has more faces 20 to the.
The regular dodecahedron is the third in an infinite set of truncated trapezohedra which can be constructed by truncating the two axial vertices of a pentagonal trapezohedron. The area a and the volume v of a truncated dodecahedron of. A dodecahedron which looks exactly the same from all faces is also a platonic solid. The shape is the 4d analogue of a dodecahedron, composed of 120 truncated dodecahedra and 600 tetrahedra. Truncated dodecahedron project gutenberg selfpublishing. This is a still from andrew weimholts pages on the 120 cell. Orthogonal projection of the 120 cell, the 4d analog of the dodecahedron. Sep 26, 2014 check out the 4dimensional analogue of the dedocahedron, the hyper dodecahedron or the 120 cell. Dodecahedron has 683 books on goodreads, and is currently reading feathers by jacqueline woodson, the heart of the buddhas teaching. In 4d all the dodecahedra are regular, but in this 3d shadow, angles are necessarily distorted, so only the innermost and outermost dodecahedra appear regular. The chamfered dodecahedron is a convex polyhedron with 80 vertices, 120 edges, and 42 faces. It can be thought of as the 4dimensional analog of the regular dodecahedron. The edgegraphs of mpolygons and of 5 platonic solids admit unique isometric embedding into a half cube of dimension m and 6, resp.
The rhombic dodecahedron has several stellations, the first of which is also a parallelohedral spacefiller. W is a torsion free subgroup of minimal index, then the corresponding hyperbolic manifold h n. Gyrochora are duals of step prisms, so use 4dcreate 4d step prism in stella4d, enter 7 2 when prompted. However, the pentagons are not constrained to be regular, and the underlying atomic arrangement has no true fivefold symmetry axis. Pictures of the polyhedra which can be used as dice, in that there is a symmetry taking any face to any other face.
Let p be the rightangled dodecahedron or 120cell in hyperbolic space, and let w be the group generated by reflections across codimensionone faces of p. If you wanted to represent a number 5 as a 2d shape youd make a pentagon, as a 3d shape youd make dodecahedron. Jan 01, 1970 the twinned passions of dodeca and hedron, told in a lighthanded minimalism that belies the sheer virulence. His pages also have foldout models of the other 4dimensional platonic solids and you can rotate them with your mouse. Let p be the rightangled hyperbolic dodecahedron or 120 cell, and let w be the group generated by reflections across codimensionone faces of p. May 22, 2007 i tried various new things, like embedding a 3d projection of a 4d 120cell into a dodecahedron, keplers pentagonbased subdivisionbased fractal faces and dodecahedral cutouts of surfaces arising from random solutions of the helmholtz equation this is an interesting topic from physics and can neatly be done using the new. Arts bathsheba the 120cell national museum of mathematics. The pyritohedron, a common crystal form in pyrite, is an irregular pentagonal dodecahedron. Dodecahedron simple english wikipedia, the free encyclopedia. Posted by madison knowe on august 1, 2016 august 1, 2016. Pdf small covers of the dodecahedron and the 120cell. Higher dimensional dodecahedra as models of the macro and.
The skeleton of the 120cell is not 5gonal sciencedirect. Authors answer 120 dodecahedron i read encyclopedias for fun. Finally, heres an di erent way to get the 120cell from the dodecahedron. Construction of the hyper dodecahedron construction of the hyper dodecahedron. A hypersphere is the threedimensional surface of a fourdimensional ball. It is also called a c120, dodecaplex short for dodecahedral complex. The 120 dodecahedra constituting the 3dimensional boundary of the 120cell can be analyzed into 9 layers containing 1, 12, 20, 12, 30, 12, 20, 12, 1 solids respctively. A 120 cell is sitting in x, y, z, wspace with one of its dodecahedral faces in the 3plane. It provided the nets the flat expression of the preconstructed solid and ways to create shapes with a protractor and a compass. A dodecahedron is a solid shape that has 12 surfaces. The stellations of the dodecahedron make up three of the four keplerpoinsot polyhedra. Alicia boole stotts first article, on certain series of sections of the regular fourdimensional hypersolids, published in 1900, is the basis for this months feature column.
Garnet cluster green 01 crystal dodecahedron 12 sided healing. Just as a dodecahedron can be built up as a model with 12 pentagons, 3 around each vertex, the dodecaplex can be built up from 120 dodecahedra, with 3 around each edge. These projections are also used in showing the fourdimensional 120 cell, a regular 4dimensional polytope, constructed from 120 dodecahedra, projecting it down to 3dimensions. Heres the 4dimensional analogue of the dodecahedron, projected into 3space for the occasion of this sculpture. It is made of 120 truncated dodecahedral and 600 tetrahedral cells. This website uses cookies to improve your experience while you navigate through the website. Citeseerx small covers of the dodecahedron and the 120cell.
Citeseerx document details isaac councill, lee giles, pradeep teregowda. In geometry, the 120cell is the convex regular 4polytope with schlafli symbol 5,3,3. In fact three of the four keplerpoinsot solids are facetings of the icosahedron. In geometry, the 120cell is the convex regular 4polytope with schlafli symbol 5, 3,3. Buy garnet cluster green 01 crystal dodecahedron 12 sided healing stone display specimen 1. Orthogonal projection of the 600 cell, the 4d analog of the icosahedron. The thinner the line, the further it is away in 4 dimensions.