Generation of an icosahedron by the intersection of five tetrahedra: geometrical and crystallographic features of the intermediate polyhedra

Livio Zefiro
Dip.Te.Ris, Universita' di Genova, Italy



Notes
  • tested with Internet Explorer 6, Mozilla Firefox 2 and Opera 9.22 at 1024x768 and 1280x1024 pixels
  • many of the images have been realized by SHAPE 7.0, a software program for drawing the morphology of crystals, distributed by Shape Software
  • hovering with the mouse on the images, sometimes the symbol of a pointer appears: if your web browser is set up to visualize VRML (Virtual Reality Modeling Language) files, by clicking with the left button of the mouse it should be possible to visualize in a new window the corresponding dynamic image, that can be enlarged, rotated, shifted...
    A good Web3D visualizer can be downloaded from here.
TABLE of CONTENTS
(one can jump directly to each item by clicking on it)
  • Introduction
  • From tetrahedron to icosahedron
  • Composition of 2 tetrahedra
  • Composition of 3 tetrahedra
  • Composition of 4 tetrahedra
  • Composition of 5 tetrahedra
  • Acknowledgements
  • References
  • Links

  • Introduction

        It is well known that an icosahedron can be decomposed into five groups of four faces such that the faces belonging to each group have the same orientation of the faces of a tetrahedron.
        The reciprocal procedure implies that, starting from a tetrahedron, the stepwise addition of other four tetrahedra, properly orientated, leads to icosahedron, through the intermediate polyhedra obtained by the intersection of compound polyhedra made of two, three and four tetrahedra, respectively.
        The orientation of each compound polyhedron made of n tetrahedra depends on the orientation of the constituting tetrahedra, but one can recognize, by proper rotations, the congruence of all the 5!/[(5-n)! n!] compound polyhedra made of n tetrahedra. The same thing occurs with their intersections, so that, after iso-orientation, the polyhedra intermediate between tetrahedron (43m crystallographic point group) and icosahedron (m35 point group) can be compared and their symmetry easily identified: whereas in the intersection of both two and three tetrahedra the symmetry is trigonal (32 point group), in the intersection of four tetrahedra the symmetry is cubic (23 point group).

    (clicking on the following images one can visualize the corresponding VRML files)
    The three intermediate forms between tetrahedron (43mpoint group) and icosahedron (m35 point group) are made up of:
  • a trigonal trapezohedron (6 brown faces) and a pinacoid (2 fuchsia faces): 32 trigonal point group
  • a trigonal trapezohedron (6 brown faces) and a rhombohedron (6 fuchsia faces): 32 trigonal point group
  • a pentagon-dodecahedron (12 brown faces) and a tetrahedron (4 fuchsia faces): 23 cubic point group

  •     Due to the lack of mirrors and inversion centre, such intermediate polyhedra can exist in enantiomorphic (or chiral) forms, whereas the icosahedron, with its fifteen mirrors and an inversion centre, admits only the possibility of a chiral colouring. On the other hand, enantiomorphism characterizes all the compound polyhedra made of tetrahedra, included the one made of five tetrahedra.
       A particular interest is incidental to the polyhedron corresponding to the intersection of four tetrahedra, named tetrahedrally stellated icosahedron by George W. Hart; its dual, consisting in the truncation of four out of twenty vertices of a regular dodecahedron, can be analogously named tetrahedrally truncated dodecahedron.
    (clicking on the following images one can visualize the corresponding VRML files)
    Tetrahedrally stellated icosahedron (on the left) and its dual, the tetrahedrally truncated dodecahedron (on the right). Also the axes of rotation (three 2-fold and four 3-fold axes), common to both polyhedra, are reported.


    From tetrahedron to icosahedron

        The five convex regular polyhedra (or Platonic solids), divided into polyhedra with cubic symmetry (cube, tetrahedron and octahedron) and icosahedral symmetry (icosahedron and dodecahedron), can be correlated in different ways:
     
  • Each is inscribable in another, according, for istance, to the following sequence:
    icosahedron - octahedron - tetrahedron - cube - dodecahedron


    From left to right:
  • icosahedron inscribed in an octahedron
  • octahedron inscribed in a tetrahedron
  • tetrahedron inscribed in a cube
  • cube inscribed in a dodecahedron

  •     Also, the polyhedra belonging to the dual couples cube-octahedron and dodecahedron-icosahedron are reciprocally inscribable, This is a consequence of their duality; the fifth Platonic solid, the tetrahedron, is inscribable in another tetrahedron, being autodual.
        Concerning the other combinations of polyhedra, it is worth mentioning both the octahedron inscribed in dodecahedron and the icosahedron inscribed in cube.

     Left column:
  • octahedron inscribed in cube
  • cube inscribed in octahedron
  •  Central column:
  • dodecahedron in icosahedron
  • icosahedron in dodecahedron
  •  Right column:
  • octahedron in dodecahedron
  • icosahedron in cube

  • Using the pleasing notation proposed by John Conway, tetrahedron is linked by the operators named join, ambo, gyro and snub to the other four Platonic solids (cube, octahedron, dodecahedron and icosahedron, respectively).

    Conway operators linking the tetrahedron to the other four Platonic solids.

  • A further relation between the two platonic solids having the least and the maximum number of faces is given from the possibily to generate an icosahedron by the intersection of five tetrahedra having their faces orientated like the icosahedral ones.

  • Animated gif showing the orientation of the five tetrahedra with rispect to icosahedron and, step by step, the result of their addition leading to a 5-colored icosahedron.

    The procedure leading from tetrahedron to icosahedron (already described here) can be summarized as follows.
        The starting point is a tetrahedron whose faces have the Miller indices: (111), (111), (1 1 1), (1 1 1).
    The choice, as second tetrahedron, of the one whose faces are characterized by the indices: (1 1 1), (111), (111), (111), hence placed centrosymmetrically with respect to the faces of the first tetrahedron, would lead to a compound polyhedron named stella octangula by Kepler. It can be regarded as the stellation of the octahedron resulting from the intersection of the two previous tetrahedra.

    (clicking on the following images one can visualize the corresponding VRML files)
    Views, along three different directions, of stella octangula (upper row) and octahedron (lower row) resulting from the intersection of two centrosymmetrically orientated tetrahedra.

        Such choice of the second tetrahedron is not suitable to reconstruct an icosahedron from tetrahedra, since the remaining twelve faces of the icosahedron could not be distributed among the three other tetrahedra with faces orientated as the icosahedral ones. Nevertheless, each of the four other tetrahedra that are fit to reconstruct the icosahedron can be derived just from this "forbidden" tetrahedron by a rotation of 44.48 around the direction perpendicular to one of its faces: this implies that the other three faces of each tetrahedron, making an angle of 109.46 with the first face, undergo a rotation of 41.81 (corresponding to the angle between the perpendiculars to every pair of contiguous icosahedral faces).

        In the following image one can see, in their own orientations, the four tetrahedra obtained, by the described procedure, starting from the "forbidden" grey tetrahedron (in the centre); their faces, and the faces of the red tetrahedron, result to be orientated like the twenty icosahedral faces.

    Four coloured tetrahedra, in their own orientation, obtained from the "forbidden" one (in grey, at the centre of the image): with the addition of the red tetrahedron, their intersection corresponds to an icosahedron.

    The next view of the icosahedron along the [111] direction evidences the yellow faces involved in the rotation of the "forbidden" tetrahedron.
    (clicking on the following image one can visualize the corresponding VRML file)
    Animated view of an icosahedron along the [111] axis (perpendicular to a face deriving from the red tetrahedron). The dark-grey and yellow icosahedral faces are involved in the procedure, described in the text, that has been applied in order to orientate properly the second tetrahedron; each icosahedral face derived from the yellow tetrahedron can be obtained also by a rotation of 41.81 of the contiguous dark-grey face, derived from the "forbidden" tetrahedron, around the edge shared by the two faces.

        There is an alternative method that permits to obtain a second tetrahedra starting from the red one. The stereographic projection of the rotation axes of icosahedron, described in a previous work concerning icosahedral polyhedra, lets one ascertain that there are three 2-fold axes perpendicular to every 3-fold axis present in icosahedral forms. In particular, the three 2-fold axes perpendicular to the [111] axis (and therefore parallel to the [111] face of the red tetrahedron) have the irrational indices: [1 -1/τ τ], [-1/τ τ 1], 1 -1/τ ] where τ = (√5+1)/2 =1.61803... denotes the golden ratio: by rotating the faces of red tetrahedron around each of the 2-fold axes, one directly obtains the faces of the yellow tetrahedron.
    It must be pointed out that these 2-fold axes are not symmetry operators of tetrahedron, just as it happens during the process of crystal twinning.

    a) Stereographic projection of the icosahedral rotation axes, including the drawing of the zones of 3-fold axes: three 2-fold axes belong to each zone.
    b) view along [001] of the polyhedron resulting from the intersection of {111} red and {111} yellow tetrahedra, reporting also the three 2-fold axes perpendicular to the 3-fold axis directed along [111]. The relative VRML image is visible clicking here.

    In the next table one can see that each 2-fold axis of rotation correlates four different pairs of faces made of a red and a yellow face.

    Directions of 2-fold axes correlating pairs of faces belonging to red and yellow tetrahedra
      (1 1 1) (1 1 1) (1 1 1) (1 1 1)
    (τ  -1/τ  0) [-1/τ  τ 1] [1 -1/τ  τ] [τ  1  -1/τ] -
    (0  τ  -1/τ) [τ  1 -1/τ] [-1/τ  τ 1] [1  -1/τ  τ] -
    (-1/τ  0 τ ) [1 -1/τ  τ] [τ  1 -1/τ] [-1/τ  τ  1] -
    (1 1 1) - - - [-1/τ  τ 1]
    [1 -1/τ  τ]
    [τ  1 -1/τ]

    Views, along different directions, both of the compound polyhedron made of red and yellow tetrahedra (make a comparison with stella octangula) and the polyhedron resulting from the intersection of the two tetrahedra.

        The following tables show the single steps leading from tetrahedron to icosahedron through intermediate polyhedra deriving from the intersection of two, three and four polyhedra.
        In the first table of the series one can see the ten polyhedra deriving from all the possible pairs of tetrahedra: also the indices of all the tetrahedral faces are reported.

    (clicking on the images one can visualize the corresponding VRML files)
    Congruent polyhedra corresponding to the intersection of pairs of tetrahedra that originate from the decomposition of an icosahedron into five tetrahedra
       

    1st tetrahedron

     

    Indices
    of each
    tetrahedral
    face

    (1 1 1)
    ( τ 1/τ  0)
    (0 τ  -1/τ)
    (1/τ  0 τ)

    ( 1 1 1)
    (τ 1/τ 0)
    (0 τ  1/τ)
    (-1/τ  0 τ )

    (1 1 1)
    ( τ -1/τ  0)
     (0 τ  1/τ )
    (1/τ  0 τ )

    1 1 1)
     (τ -1/τ  0)
     (0 τ -1/τ)
     (-1/τ  0 τ)

    (1 1 1)
    (1 1 1)
    (1 1 1)
    (1 1 1)

    2nd

    t
    e
    t
    r
    a
    h
    e
    d
    r
    o
    n

     (1 1 1)
     (1 1 1)
     (1 1 1)
     (1 1 1)

     ( 1 1 1 )
     (τ  -1/τ  0)
     (0  τ -1/τ)
     (-1/τ  0 τ)

     (1 1 1)
     ( τ  -1/τ  0)
     (0 τ  1/τ )
     (1/τ  0 τ )

     ( 1 1 1)
     (τ 1/τ  0)
     (0  τ  1/τ)
     (-1/τ  0  τ )

     (1 1 1)
     ( τ 1/τ  0)
     (0 τ  -1/τ)
     
    (1/τ  0 τ)

        It is not difficult to realize that the previous polyhedra, made of pair of tetrahedra, are all congruent (regardless of colouring), as visualized by the animated gif images contained in the following table relative to compound polyhedra, deriving from the combination of the red tetrahedron with the four others, and their intersections: starting from every initial orientation of the different couples of tetrahedra, by appropriate rotations a final common orientation is obtained in every case.


    In the same way, ten congruent sets of three tetrahedra can be obtained adding a third tetrahedron to the previous pairs of tetrahedra.
    (clicking on the images one can visualize the corresponding VRML files)
    Congruent intersections of three tetrahedra obtained by the addition of a third tetrahedron to each pair of tetrahedra

    The next step shows how the five congruent sets of four tetrahedra can be obtained from the ten sets of three tetrahedra by the addition of a fourth tetrahedron.

    (clicking on the images one can visualize the corresponding VRML files)

    Congruent intersections of four tetrahedra obtained by the addition of a fourth tetrahedron to each set of three tetrahedra

     


     


     


     


     


    The last step consists in the addition of a fifth tetrahedron to each set of four tetrahedra, giving as result a 5-coloured icosahedron.
    (clicking on the images one can visualize the corresponding VRML files)

    Addition of a fifth tetrahedron to each set of four tetrahedra leading to icosahedron

    + + + + +


        To summarize, in the following table both the compound polyhedra and the result of the intersections of tetrahedra are shown, starting from the red tetrahedron with the successive addition, step by step, of the yellow, green, pale-blue and violet tetrahedra.
    (clicking on the images, one can visualize the corresponding VRLM files)




    Pairs of images relative to the compound polyhedra, made of 2-3-4-5 tetrahedra (on the left), and their intersections (on the right), leading from tetrahedron to icosahedron; viewing direction, scale and orientation of the polyhedra are the same for all the images.


    Compound polyhedron made of 2 tetrahedra and solid generated by their intersection

    The following animated GIF, consisting of four frames, shows sequentially the images along different directions, included 2-fold and 3-fold rotation axes, relative to the pairs of:
  • chiral compound polyhedra made of red and yellow tetrahedra (first row)
  • chiral solids generated by the intersection of the two tetrahedra (second row)
    The polyhedra in the second column have been properly orientated to evidence their mirror symmetry (concerning both shape and colouring) with respect to the polyhedra in the first column.

  • Animated GIF of chiral compounds of two tetrahedra and their intersections, seen by ortographic viewing and along 2-fold and 3-fold axes.

    Starting from the initial position a), the solid generated by the intersection of red and yellow tetrahedra is shown in b) after an appropriate rotation, in order to evidence the presence of a vertical 3-fold axis and three 2-fold axes perpendicular to the 3-fold axis, characterizing the 32 point group. In c) the same colour has been given to all the faces belonging to each of the two crystallographic forms: trigonal trapezohedron (6 brown faces) and pinacoid (2 fuchsia faces).


  • Left: view, along the [001] direction, of the faces of the polyhedron corresponding to the intersection of two tetrahedra and relative stereographic projection; in conformity with the 32 point group, the forms are a trigonal trapezohedron and a pinacoid (here and in all the following stereographic projections, circles indicate the positions of faces belonging to the North hemisphere, crosses indicate faces belonging to the South hemisphere)
  • Right: orthographic view of the only trigonal trapezohedron.

  • Three alternative colourings of the faces of the polyhedron resulting from the intersection of two tetrahedra. The first two colourings refer to:
  • the faces deriving from red and yellow tetrahedra
  • the trigonal trapezohedron and the pinacoid constituting the polyhedron
    In order to evidence the single faces, the sharing of edges by faces having equal colouring should be avoided; since the triangular faces share their edges with three pentagonal faces, at least four colours need to be used (
    third image): one for the two triangular faces and three other colours for the two sets of three pentagonal faces related by the 3-fold axis. Moreover each pentagonal face of a set of three faces is related by two 2-fold axes to two pentagonal faces, coloured by the other two colours, belonging to the second set of three faces. Therefore, in case of this 4-coloured polyhedon, the relative symmetry results to be 3'2', where 3' and 2' indicates chromatic 3-fold and 2-fold axes, respectively.

  • The geometric features of the pentagonal faces are visible clicking here: the side lengths are relative to an unit distance of the face from the centre of the polyhedron.

    4-coloured planar nets of the chiral polyhedra deriving from the intersection of two tetrahedra: the alternative nets, in which one colour corresponds to each crystallographic form (trapezohedron or pinacoid), can be seen clicking here; a further pair of chiral nets, coloured accordingly to the red and yellow intersecting tetrahedra, is shown here.


    Compound polyhedron made of 3 tetrahedra and solid generated by their intersection

        The following animated GIF, consisting of four frames, shows sequentially the images along different directions, included 2-fold and 3-fold rotation axes, relative to the pairs of:
  • chiral compound polyhedra made of red, yellow and green tetrahedra (first row)
  • chiral solids generated by the intersection of the three tetrahedra (second row)
    The polyhedra in the second column have been properly orientated to evidence their mirror symmetry (concerning both shape and colouring) with respect to the polyhedra in the first column.

  • Animated GIF of chiral compounds of three tetrahedra and their intersections, seen by ortographic viewing and along 2-fold and 3-fold axes.

    Starting from the initial position a), the solid generated by the intersection of red, yellow and green tetrahedra is shown in b) after an appropriate rotation, in order to evidence the presence of both a vertical 3-fold axis and three 2-fold axes, perpendicular to the 3-fold axis, characterizing the 32 point group. In c) the same colour has been assigned to all the faces belonging to each of the two crystallographic forms: trigonal trapezohedron (6 brown faces) and rhombohedron (6 fuchsia faces).

    View along the [001] direction of the faces of the polyhedron resulting from the intersection of three tetrahedra and relative stereographic projection: in conformity with the 32 point group, the forms present are a trigonal trapezohedron (6 brown faces) and a rhombohedron (6 fuchsia faces).

    Orthographic views of trigonal trapezohedron (left) and rhombohedron (right) constituting the polyhedron that derives from the intersection of three tetrahedra.

    Clicking here, one can see the geometry of the quadrilateral and pentagonal faces characterizing the polyhedron that derives from the intersection of three tetrahedra; the side lengths are relative to a unit distance of the faces from the centre of the polyhedron.

    Planar nets of the chiral polyhedra deriving from the intersection of red, yellow and green tetrahedra: clicking here, one can see alternative nets, in which the same colour is assigned to all the faces belonging to to each crystallographic form (trapezohedron or rhombohedron).


    Compound polyhedron made of 4 tetrahedra and solid generated by their intersection

    The following animated GIF, consisting of three frames, shows sequentially the images along different directions, included 2-fold and 3-fold rotation axes, relative to the pairs of:
  • chiral compound polyhedra made of red, yellow, green and pale-blue tetrahedra (first row)
  • chiral solids generated by the intersection of the four tetrahedra (second row)
    The polyhedra in the second column have been properly orientated to evidence their mirror symmetry (concerning both shape and colouring) with respect to the polyhedra of the first column.

  • Animated GIF of chiral compounds of four tetrahedra and their intersections seen by orthographic viewing and along 3-fold and 2-fold axes.

    Starting from the initial position a), the solid generated by the intersection of four tetrahedra is shown in b) after an appropriate rotation, in order to evidence the presence of four three-fold axis and three perpendicular two-fold axes, characterizing the 23 point group. In c) the same colouring has been assigned to all the faces belonging to each of the two crystallographic forms: pentagon-dodecahedron (12 brown faces) and tetrahedron (4 fuchsia faces).

    View along the [001] direction and stereographic projection of the faces of the polyhedron resulting from the intersection of four tetrahedra.

        As well as by the intersection of four tetrahedra, the same polyhedron can be obtained, of course, eliminating from an icosahedron four faces, orientated like the faces of a tetrahedron (the one with {111} indices, for example). This may be achieved by extending the three triangular faces contiguous to each face of the chosen tetrahedron, until they meet in a new vertex: consequently the shape of twelve faces (out of twenty) changes from triangular to deltoidal (or kite-shaped quadrilateral), whereas the remaining four faces, centrosymmmetric with respect to the four faces eliminated, retain their triangular shape.

    Animated GIF showing the stellation process relative to four out of twenty faces of the icosahedron, leading to the tetrahedrally stellated icosahedron.

    Four 3-fold axis are parallel to the directions connecting each new vertex with the centre of the opposite triangular face, whereas three 2-fold axes are perpendicular to the midpoint of the long sides of the deltoids not shared with the triangular faces: the overall resulting symmetry corresponds to the crystallographic 23 point group.
    Consequently, the sixteen faces can be grouped in two forms: a {τ 1/τ  0} pentagon-dodecahedron and a {1 1 1} tetrahedron. (Due to the lack of both inversion centre and any mirror, also an enantiomorphic polyhedron exists, made of the same {τ 1/τ  0} pentagon-dodecahedron and a {111} tetrahedron).

    Orthographic views of the forms in which the solid generated by the intersection of four tetrahedra can be decomposed: a {τ 1/τ  0} pentagon-dodecahedron and a { 1 1 1} tetrahedron.

    Taking into account the procedure followed to get the polyhedron, George W. Hart proposed for it the name: tetrahedrally stellated icosahedron.
        To summarize, the polyhedron consists of sixteen faces and sixteen vertices: the twelve original vertices of icosahedron plus the four new vertices (of course, the edges are thirty, from the known rule E = F+V-2, valid for convex polyhedra). This circumstance implies that also its dual has the same number of faces, vertices and edges; therefore the two polyhedra are topologically identical, but not geometrically: in fact the faces of the dual consist of four equilateral triangles and twelve isosceles trapezoids (or trapezia).
        The faces of the dual can be grouped according to two crystallographic forms, coexisting in the 23 point group: a {111} tetrahedron and a {1τ 0} pentagon-dodecahedron, geometrically identical to a regular dodecahedron. Consequently, the dual of a tetrahedrally stellated icosahedron can be obtained truncating a dodecahedron by a tetrahedron, till the twelve new vertices of the truncated dodecahedron coincide with twelve of the original twenty vertices of the dodecahedron (in total, due to the truncation process, the vertices from 20 become at first (20 - 4) + (3 x 4) = 28 and, at the end, 20 - 4 = 16).
      Tetrahedrally truncated dodecahedron is the appropriate name, obviously correlated with the name of its dual, given from George W. Hart to this polyhedron.

    Animated GIF showing the truncation of four vertices of a dodecahedron, leading to the tetrahedrally truncated dodecahedron.

    View, along the [001] direction, of the faces of the tetrahedrally truncated dodecahedron and relative stereographic projection.

  • First row: chiral couple made of tetrahedrally stellated icosahedra
  • Second row: chiral couple made of tetrahedrally truncated dodecahedra
  • First column: dual couple made of tetrahedrally stellated icosahedra and tetrahedrally truncated dodecahedra
  • Second column: dual couple made of chiral tetrahedrally stellated icosahedra and chiral tetrahedrally truncated dodecahedra
    Every polyhedron includes 16 faces, 16 vertices and 30 edges and belongs to 23 point group, being characterized by three 2-fold axes and four 3-fold axes.

  •     A further interesting feature concerning the canonicalization of the previous dual couple of polyhedra has been pointed out by George W. Hart: the canonical forms of tetrahedrally stellated icosahedron and tetrahedrally truncated dodecahedron are geometrically self-dual.
    The kite-shaped faces of the canonical form have edges in the ratio 1: 0.803 and the angle between the shorter sides is 111.15, whereas the other 3 angles are equal (82.95).

    Geometry of kite-shaped faces of the tetrahedrally stellated icosahedron (left), trapezoidal faces of the tetrahedrally truncated dodecahedron (centre), kite-shaped faces of the related canonical form (right).

    (clicking on the following images one can visualize the VRML file relative to the canonical form)
    View, along a 2-fold axis (left) and a 3-fold axes (right), of a polyhedron approximating rather closely the canonical form of both the tetrahedrally stellated icosahedron and the chiral tetrahedrally truncated dodecahedron.

    Planar nets of chiral tetrahedrally stellated icosahedra: their four colours remind one that the polyhedra derive from the intersection of red, yellow, green and pale-blue tetrahedra; alternative nets, in which one colour corresponds to each crystallographic form (pentagon-dodecahedron or tetrahedron), can be seen clicking here.

     Nets of a chiral couple made of tetrahedrally truncated dodecahedra, duals of the chiral couple made of tetrahedrally stellated icosahedra.
    The isosceles trapezia have the following remarkable features: the length of the shorter parallel side is equal to the two not parallel sides and the value of the ratio between the parallel sides corresponds to the golden ratio
    τ = (√5+1)/2 =1.61803...


    Nets of the chiral canonical forms of tetrahedrally stellated icosahedron and tetrahedrally truncated dodecahedron.


    Compound polyhedron made of 5 tetrahedra and icosahedron generated by their intersection

        The following animated GIF, consisting of three frames, shows sequentially the images along different directions, included 5-fold and 3-fold rotation axes, relative to the pairs of:
  • chiral compound polyhedra made of red, yellow, green, pale-blue and violet tetrahedra (first row)
  • chiral colouring of the solids generated by the intersection of the five tetrahedra (second row)
    The polyhedra in the second column have been properly orientated to evidence their mirror colouring relatively to the polyhedra of the first column; in the instance of the compound polyhedra, the mirror symmetry concerns also their shape.

  • Animated GIF of chiral compounds of five tetrahedra and their intersections seen by orthographic viewing and along 5-fold and 3-fold axes.


    Views, along the [001] direction, of the chiral colouring of the icosahedron, whereas the stereographic projection is unique; in fact, the presence of both fifteen mirrors and inversion centre rules out the possibility of enantiomorphism concerning the geometrical shape of icosahedron itself.

    Compound polyhedra made of 5 tetrahedra
    view along [0 1 τ] view along [1 τ0] view along [τ0 1]
    view along [0 1 τ ] view along [1 τ0] view along [τ0 1 ]
    view along [0 1 τ] view along [1 τ0] view along [τ0 1 ]
    view along [0 1 τ ] view along [1 τ0] view along [τ0 1]
    Chiral compound polyhedra made of 5 tetrahedra
    view along [0 1 τ ] view along [1 τ0] view along [τ0 1 ]
    view along [0 1 τ] view along [1 τ0] view along [τ0 1]
    view along [0 1 τ ] view along [1 τ0] view along [τ0 1]
    view along [0 1 τ] view along [1 τ0] view along [τ0 1 ]
    Views, along all the directions corresponding to 5-fold axes, of the two chiral compound polyhedra made of five tetrahedra.
    In every row there are three pairs of views, along [uvw] and [u v w] centrosymmetric directions, showing that both colour pattern and geometric shape of chiral compound polyhedra are symmetric with respect to an intermediate vertical mirror (compare the images of 1st and 4th, 2nd and 5th, 3rd and 6th columns).


    First row: chiral compound polyhedra made of five tetrahedra, whose vertices coincide with the vertices of a dodecahedron; their enantiomorphism concerns both colouring and shape of the polyhedra.
    Second row: chiral icosahedra resulting from the intersection of the previous compound polyhedra, positioned with respect to the outer dodecahedron (also the position of the vertices of each visible coloured tetrahedron has been reported); they are enantiomorphic only concerning the colouring of the triangular faces.

        As shown in the following image, an interesting dissection of the icosahedron generates two pentagonal pyramids and a pentagonal antiprism (or para-bidiminished icosahedron, according to Johnson's systematic nomenclature): it suggests, among the 43380 possible different nets of the icosahedron (Buekenhout, F. and Parker, M., 1998), the choice of a net including a chain of ten side-sharing triangles; two other sets of five vertex-sharing triangles, placed on the opposite sides of the chain, are connected to it.
    Dissection of an icosahedron into two pentagonal pyramids and a pentagonal antiprism.

    Chiral 5-coloured nets of icosahedron.


    Acknowledgements

    Many thanks are due to Maria Rosa Ardigo' and Riccardo Basso for the fruitful discussions and to Fabio Somenzi for his precious help concerning the translation of the text.


    References

    H. Martyn Cundy and A. P. Rollet
        "Mathematical Models"
    , Oxford University Press, 1954

    Peter R. Cromwell
        "Polyhedra"
    , Cambridge University Press, 1997

    F. Buekenhout and M. Parker
        "The Number of Nets of the Regular Convex Polytopes in Dimension Less or Equal to 4", Disc. Math., 186, 69-94, 1998

    Norman W. Johnson
        "Convex Polyhedra with Regular Faces", Canadian Journal of Mathematics, 18, 169-200, 1966.


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