/*
    Copyright 2000, University of Washington

    Permission to use, copy, modify, and distribute this software and its
    documentation for any purpose and without fee is hereby granted,
    provided that the above copyright notice appear in all copies and that
    both the copyright notice and this permission notice and warranty
    disclaimer appear in supporting documentation, and that the names of
    the authors or their employers not be used in advertising or publicity
    pertaining to distribution of the software without specific, written
    prior permission.

    The authors and their employers disclaim all warranties with regard to
    this software, including all implied warranties of merchantability and
    fitness.  In no event shall the authors or their employers be liable
    for any special, indirect or consequential damages or any damages
    whatsoever resulting from loss of use, data or profits, whether in an
    action of contract, negligence or other tortious action, arising out of
    or in connection with the use or performance of this software.
*/

/*
    isohedral.ih

    This file contains the set of information used by our software 
    to encode the isohedral tilings.  The only element missing is the
    set of tiling vertex parameterizations, which can be found in 
    params.py.

    The format of this file is slightly different than what's described
    in the paper (the paper's version is altered to simplify the exposition).
    Here are the important differences:

    * Some incidence symbols are _rotated_ with respect to the 
      presentation in Grunbaum and Shephard.  That means that we 
      start the enumeration of edges at a different tiling edge, 
      resulting in a different (but equivalent) symbol.  I did this
      in the cases that the incidence symbols weren't compatible with
      their topological types.

    * The twelve IH types that can only be represented with marked
      tiles are not given (see figure 6.2.5 of G+S).  These tiles are
      uninteresting from an ornamental point of view, since their edges
      cannot be modified.

    * Indices are zero-based, not one-based.  This affects the 
      assignment of colours to indices, the two colour permutations,
      and the edge numbers given in the rules section.

    * Aspects have explicit names, not just indices.  When a rule 
      refers to an aspect, it refers to that aspect by name.
    
    * The format for rules is a little more rich.  Instead of always 
      starting from the first aspect in the translational unit at
      the origin, you can start from _any_ aspect by naming it.  Thus
      the rule

          aspect 2r 1r:2,3
    
      means "to obtain the transform for aspect "2r", start from aspect
      "1r", hop across the edge in that tile with index 2, then across
      the edge of the next tile with index 3."  Each hop multiplies on
      a transform matrix.  Each hop is taken from the tile arrived at 
      via the previous hop, and edge indices (zero-based) are measured
      from the start of the incidence symbol.


    Please contact me at csk@cs.washington.edu if you have questions
    about the information in this file.

    For more details about our system, see the paper "Escherization",
    by Craig S. Kaplan and David H. Salesin, in the SIGGRAPH 2000 
    Conference Proceedings.

    Craig S. Kaplan
*/

/*
    December 1st, 2000

        Many of the colourings have been updated.  There was a bug in
        my implementation that reversed translation vectors, and this
        file was correct relative to that bug.  It should be fixed now.
        Big thanks to Jens Kilian for finding this problem.

        Note: there are probably still implementation problems related to 
        dependencies in the order that aspect rules are computed.  Use a 
        topological sort.

        Note: at some point, provide perfect colourings.
*/

/* 
    May 8th, 2002

        Recently, I noticed a problem in the way the tiling vertex
        parameterizations interact with the incidence symbols given
        here.  There should be no reason why you can't use the
        vertex parameterizations to create tile shapes with degenerate
        tiling edges (i.e., collapse two tiling edges of the hexagon
        in IH1 down to zero, and obtain a parallelogram).  The
        problem is that the computation of aspect transforms or 
        translation vectors can get messed up with you allow 
        degenerate edges, since you then can't determine the 
        rotation or reflection across that edge.

        I don't have a really good solution at this time.  The 
        answer might be to offer alternate versions of some tiling
        types that avoid the bug.  For example, because aspect "2"
        of IH27 is computed by jumping across edge 0, that edge
        can't be degenerate.  You need an alternate IH27 that doesn't
        use edge 0 to get the transform for aspect "2".  On the 
        other hand, it might turn out that the correct transform
        across a degenerate edge is _always_ either a 180 degree
        rotation or a reflection across an easily-determined line.
        I'll think about this problem some more.
*/

template IH01 {
    symbol [a+b+c+d+e+f+;d+e+f+a+b+c+]
    topology 3^6
    aspects 1
    colouring 3 (0) (1 2 0) (2 0 1)
    rules
        translate T1 1:0
        translate T2 1:1
}

template IH02 {
    symbol [a+b+c+d+e+f+;b-a-f+e-d-c+]
    topology 3^6
    aspects 1,1r
    colouring 3 (0 1) (1 2 0) (0 1 2)
    rules
        aspect 1r 1:0
        translate T1 1:2
        translate T2 1r:3
}

template IH03 {
    symbol [a+b+c+d+e+f+;c-e+a-f-b+d-]
    topology 3^6
    aspects 1,1r
    colouring 3 (0 1) (2 0 1) (2 0 1)
    rules
        aspect 1r 1:0
        translate T1 1:1
        translate T2 1r:5
}

template IH04 {
    symbol [a+b+c+d+e+f+;a+e+c+d+b+f+]
    topology 3^6
    aspects 1,2
    colouring 3 (0 1) (2 0 1) (2 0 1)
    rules
        aspect 2 1:0
        translate T1 1:1
        translate T2 2:3
}

template IH05 {
    symbol [a+b+c+d+e+f+;a+e+d-c-b+f+]
    topology 3^6
    aspects 1,2,1r,2r
    colouring 3 (0 1 2 1) (2 0 1) (2 0 1)
    rules
        aspect 2 1:0
        aspect 1r 1:3
        aspect 2r 1r:5
        translate T1 1:1
        translate T2 1:3,5,2,0
}

template IH06 {
    symbol [a+b+c+d+e+f+;a+e-c+f-b-d-]
    topology 3^6
    aspects 1,2,1r,2r
    colouring 3 (0 1 2 2) (1 2 0) (0 1 2)
    rules
        aspect 2 1:0
        aspect 1r 1:1
        aspect 2r 2:1
        translate T1 1r:1
        translate T2 2r:5
}

template IH07 {
    symbol [a+b+c+d+e+f+;b+a+d+c+f+e+]
    topology 3^6
    aspects 1,2,3
    colouring 3 (0 1 2 ) (0 1 2) (0 1 2)
    rules
        aspect 2 1:0
        aspect 3 1:1
        translate T1 2:3
        translate T2 3:4
}

template IH08 {
    symbol [a+b+c+a+b+c+;a+b+c+]
    topology 3^6
    aspects 1
    colouring 3 (0) (1 2 0) (2 0 1)
    rules 
        translate T1 1:0
        translate T2 1:1
}

template IH09 {
    symbol [a+b+c+a+b+c+;a+c-b-]
    topology 3^6
    aspects 1,1r
    colouring 3 (0 1) (2 0 1) (0 1 2)
    rules
        aspect 1r 1:1
        translate T1 1:0
        translate T2 1r:4
}

template IH10 {
    symbol [a+b+a+b+a+b+;b+a+]
    topology 3^6
    aspects 1
    colouring 3 (0) (1 2 0) (2 0 1)
    rules
        translate T1 1:0 
        translate T2 1:5
}

template IH11 {
    symbol [a+a+a+a+a+a+;a+]
    topology 3^6
    aspects 1
    colouring 3 (0) (1 2 0) (2 0 1)
    rules
        translate T1 1:0
        translate T2 1:1
}

template IH12 {
    symbol [a b+c+d c-b-;d c-b-a]
    topology 3^6
    aspects 1
    colouring 3 (0) (1 2 0) (2 0 1)
    rules
        translate T1 1:0
        translate T2 1:1
}

template IH13 {
    symbol [a b+c+ d c-b-;d b+c+a]
    topology 3^6
    aspects 1,1r
    colouring 3 (0 1) (2 0 1) (0 1 2)
    rules 
        aspect 1r 1:1
        translate T1 1:0
        translate T2 1r:5
}

template IH14 {
    symbol [a+b+c+c-b-a-;c-b-a-]
    topology 3^6
    aspects 1
    colouring 3 (0) (1 2 0) (2 0 1)
    rules 
        translate T1 1:0
        translate T2 1:5
}

template IH15 {
    symbol [a+b+c+c-b-a-;a+b-c+]
    topology 3^6
    aspects 1,2
    colouring 3 (0 1) (2 0 1) (0 1 2)
    rules 
        aspect 2 1:0
        translate T1 1:1
        translate T2 2:2
}

template IH16 {
    symbol [a+b+c+c-b-a-;a-c+b+]
    topology 3^6
    aspects 1,2,3
    colouring 3 (0 1 2) (0 1 2) (0 1 2)
    rules
        aspect 2 1:0
        aspect 3 1:5
        translate T1 2:4
        translate T2 2:2
}

template IH17 {
    symbol [a b+b-a b+b-;a b+]
    topology 3^6
    aspects 1
    colouring 3 (0) (1 2 0) (2 0 1)
    rules
        translate T1 1:0
        translate T2 1:1
}

template IH18 {
    symbol [a b a b a b;b a]
    topology 3^6
    aspects 1
    colouring 3 (0) (1 2 0) (2 0 1)
    rules
        translate T1 1:0
        translate T2 1:1
}

// IH19 is only representable with marked templates.

template IH20 {
    symbol [a a a a a a;a]
    topology 3^6
    aspects 1
    colouring 3 (0) (1 2 0) (2 0 1)
    rules
        translate T1 1:0
        translate T2 1:1
}

template IH21 {
    // The incidence symbol here is rotated WRT the book to account
    // for the inconsistency with the topology type.
    symbol [a+b+c+d+e+;a+c+b+e+d+]
    topology 3^4.6
    aspects 1,2,3,4,5,6
    colouring 3 (0 1 0 1 0 1) (1 2 0) (2 0 1)
    rules
        aspect 2 1:4
        aspect 3 2:4
        aspect 4 3:4
        aspect 5 4:4
        aspect 6 5:4
        translate T1 3:1
        translate T2 4:0
}

template IH22 {
    // This one was rotated too.
    symbol [a+b+c+d+e+;b-a-e+d-c+]
    topology 3^3.4^2
    aspects 1,1r
    colouring 3 (0 1) (2 0 1) (0 1 2)
    rules
        aspect 1r 1:3
        translate T1 1r:0
        translate T2 1r:1
}

template IH23 {
    symbol [a+b+c+d+e+;a+b+e+d+c+]
    topology 3^3.4^2
    aspects 1,2
    colouring 3 (0 1) (1 2 0) (2 0 1)
    rules
        aspect 2 1:0
        translate T1 1:2
        translate T2 2:3
}

template IH24 {
    symbol [a+b+c+d+e+;a+b+e+d-c+]
    topology 3^3.4^2
    aspects 1,2,1r,2r
    colouring 3 (0 1 2 0) (2 0 1) (1 2 0) 
    rules
        aspect 2 1:0
        aspect 1r 2r:0
        aspect 2r 2:3
        translate T1 1:4
        translate T2 1r:3
}

template IH25 {
    symbol [a+b+c+d+e+;b-a-e+d+c+]
    topology 3^3.4^2
    aspects 1,2,1r,2r
    colouring 3 (0 1 2 0) (2 0 1) (0 1 2) 
    rules
        aspect 2 1:3
        aspect 1r 2r:3
        aspect 2r 2:1
        translate T1 1:2
        translate T2 1r:1
}

template IH26 {
    symbol [a+a-b+c b-;a+b-c]
    topology 3^3.4^2
    aspects 1,2
    colouring 3 (0 1) (1 2 0) (0 1 2) 
    rules 
        aspect 2 1:3
        translate T1 1:2
        translate T2 2:0
}

template IH27 {
    symbol [a+b+c+d+e+;a+d-e-b-c-]
    topology 3^2.4.3.4
    aspects 1,2,1r,2r
    colouring 3 (0 1 2 2) (0 1 2) (1 2 0) 
    rules
        aspect 2 1:0
        aspect 1r 1:1
        aspect 2r 2:1
        translate T1 2r:4
        translate T2 1r:1
}

template IH28 {
    symbol [a+b+c+d+e+;a+c+b+e+d+]
    topology 3^2.4.3.4
    aspects 1,2,3,4
    // After reading G.C. Shephard's "What Escher Might Have Done" in
    // _M.C. Escher: Art and Science_, I realized something interesting:
    // this colouring isn't perfect.  The paper doesn't claim that all
    // our colourings are perfect, but I was operating under the tacit
    // assumption that they were.  So it's somewhat ambiguous whether
    // Escher would have used this colouring in one of his tilings.
    // The fish on the cover of _Visions of Symmetry_ are of this tiling
    // type and use a colouring with four colours.  In fact, Shephard
    // shows that a perfect 3-colouring of this tiling type cannot exist.
    colouring 3 (0 1 2 0) (1 2 0) (0 1 2) 
    rules
        aspect 2 1:0
        aspect 3 2:2
        aspect 4 2:3
        translate T1 4:3
        translate T2 3:2
}

template IH29 {
    symbol [a b+c+c-b-;a c+b+]
    topology 3^2.4.3.4
    aspects 1,2,3,4
    colouring 3 (0 1 2 2) (1 2 0) (0 1 2)
    rules
        aspect 2 1:0
        aspect 3 1:1
        aspect 4 1:4
        translate T1 3:4
        translate T2 4:4
}

template IH30 {
    symbol [a+b+c+d+;d+b-c-a+]
    topology 3.4.6.4
    aspects 1,2,3,1r,2r,3r
    colouring 3 (0 1 2 1 2 0) (1 2 0) (2 0 1) 
    rules
        aspect 2 1:0
        aspect 3 2:0
        aspect 1r 1:1
        aspect 2r 1r:0
        aspect 3r 2r:0
        translate T1 3r:2
        translate T2 3:2,1
}

template IH31 {
    symbol [a+b+c+d+;d+c+b+a+]
    topology 3.4.6.4
    aspects 1,2,3,4,5,6
    // This tiling has a simpler but visually less appealing alternate
    // colouring:
    // colouring 3 (0 1 0 1 0 1) (2 0 1) (1 2 0)

    // I prefer this colouring:
    colouring 3 (0 1 2 0 1 2) (0 1 2) (0 1 2)
    rules
        aspect 2 1:1
        aspect 3 2:1
        aspect 4 3:1
        aspect 5 4:1
        aspect 6 5:1
        translate T1 3:0
        translate T2 5:3
}

template IH32 {
    symbol [a+b+b-a-;a-b-]
    topology 3.4.6.4
    aspects 1,2,3,4,5,6
    colouring 3 (0 2 1 1 0 2) (1 2 0) (2 0 1) 
    rules
        aspect 2 1:3
        aspect 3 2:3
        aspect 4 1:1
        aspect 5 4:3
        aspect 6 5:3
        translate T1 5:2
        translate T2 2:2,2
}

template IH33 {
    symbol [a+b+c+d+;d+c+b+a+]
    topology 3.6.3.6
    aspects 1,2,3
    colouring 3 (0 1 2) (0 1 2) (0 1 2)
    rules 
        aspect 2 1:0
        aspect 3 2:0
        translate T1 2:1
        translate T2 3:2
}

template IH34 {
    symbol [a+b+a+b+;b+a+]
    topology 3.6.3.6
    aspects 1,2,3
    colouring 3 (0 1 2) (0 1 2) (0 1 2)
    rules
        aspect 2 1:0
        aspect 3 2:0
        translate T1 2:3
        translate T2 3:0
}

// IH35 can only be realized with marked templates.

template IH36 {
    symbol [a+a-b+b-;b-a-]
    topology 3.6.3.6
    aspects 1,2,3
    colouring 3 (0 1 2) (0 1 2) (0 1 2)
    rules
        aspect 2 1:0
        aspect 3 1:3
        translate T1 2:0
        translate T2 3:2
}

template IH37 {
    symbol [a+a-a+a-;a-]
    topology 3.6.3.6
    aspects 1,2,3
    colouring 3 (0 1 2) (0 1 2) (0 1 2)
    rules
        aspect 2 1:0
        aspect 3 2:3
        translate T1 2:2
        translate T2 3:1
}

template IH38 {
    symbol [a+b+c+;c+b-a+]
    topology 3.12^2
    aspects 1,2,3,1r,2r,3r
    colouring 3 (0 2 1 1 0 2) (1 2 0) (2 0 1)
    rules
        aspect 2 1:0
        aspect 3 2:0
        aspect 1r 3r:0
        aspect 2r 2:1
        aspect 3r 2r:0
        translate T1 3r:1
        translate T2 3:1,2,1
}

template IH39 {
    symbol [a+b+c+;c+b+a+]
    topology 3.12^2
    aspects 1,2,3,4,5,6
    colouring 3 (0 2 1 0 1 2) (2 0 1) (1 2 0)
    rules 
        aspect 2 1:0
        aspect 3 2:0
        aspect 4 2:1
        aspect 5 4:0
        aspect 6 5:0
        translate T1 3:1,0,1
        translate T2 6:1
}

template IH40 {
    symbol [a+b a-;a-b]
    topology 3.12^2
    aspects 1,2,3,4,5,6
    colouring 3 (1 0 2 2 0 1) (1 2 0) (2 0 1)
    rules
        aspect 2 1:2
        aspect 3 2:2
        aspect 4 2:1
        aspect 5 4:2
        aspect 6 5:2
        translate T1 6:1
        translate T2 5:1,2
}

template IH41 {
    symbol [a+b+c+d+;c+d+a+b+]
    topology 4^4
    aspects 1
    colouring 2 (0) (1 0) (1 0)
    rules
        translate T1 1:0
        translate T2 1:1
}

template IH42 {
    symbol [a+b+c+d+;c+b-a+d-]
    topology 4^4
    aspects 1,1r
    colouring 2 (0 1) (1 0) (0 1)
    rules
        aspect 1r 1:1
        translate T1 1:0
        translate T2 1r:3
}

template IH43 {
    symbol [a+b+c+d+;c-d+a-b+]
    topology 4^4
    aspects 1,1r
    colouring 2 (0 1) (1 0) (0 1)
    rules
        aspect 1r 1:0
        translate T1 1:1
        translate T2 1r:0
}

template IH44 {
    symbol [a+b+c+d+;b-a-d-c-]
    topology 4^4
    aspects 1,1r
    colouring 2 (0 1) (0 1) (0 1)
    rules
        aspect 1r 1:0
        translate T1 1r:0
        translate T2 1r:2
}

template IH45 {
    symbol [a+b+c+d+;c-b-a-d-]
    topology 4^4
    aspects 1,1r
    colouring 2 (0 1) (0 1) (0 1)
    rules
        aspect 1r 1:0
        translate T1 1r:0
        translate T2 1r:1
}

template IH46 {
    symbol [a+b+c+d+;a+b+c+d+]
    topology 4^4
    aspects 1,2
    colouring 2 (0 1) (0 1) (0 1)
    rules
        aspect 2 1:0
        translate T1 2:2
        translate T2 2:1
}

template IH47 {
    symbol [a+b+c+d+;c+b+a+d+]
    topology 4^4
    aspects 1,2
    colouring 2 (0 1) (1 0) (0 1)
    rules
        aspect 2 1:1
        translate T1 1:0
        translate T2 2:3
}

template IH49 {
    symbol [a+b+c+d+;a-b+c-d+]
    topology 4^4
    aspects 1,2,1r,2r
    colouring 2 (0 1 1 0) (0 1) (0 1)
    rules
        aspect 2 1:1
        aspect 1r 1:0
        aspect 2r 2:0
        translate T1 1r:2
        translate T2 2:3
}

template IH50 {
    symbol [a+b+c+d+;c+b-a+d+]
    topology 4^4
    aspects 1,2,1r,2r
    colouring 2 (0 1 1 0) (0 1) (1 0)
    rules
        aspect 2 1:3
        aspect 1r 1:1
        aspect 2r 2:1
        translate T1 2r:3,1
        translate T2 1:0
}

template IH51 {
    symbol [a+b+c+d+;c-b+a-d+]
    topology 4^4
    aspects 1,2,1r,2r
    colouring 2 (0 1 1 0) (0 1) (0 1)
    rules
        aspect 2 1:3
        aspect 1r 1:0
        aspect 2r 2:2
        translate T1 1r:0
        translate T2 2:1
}

template IH52 {
    symbol [a+b+c+d+;c-d-a-b-]
    topology 4^4
    aspects 1,2,1r,2r
    colouring 2 (0 0 1 1) (0 1) (0 1)
    rules
        aspect 2 1r:3
        aspect 1r 1:0
        aspect 2r 2:0
        translate T1 2r:1
        translate T2 1r:0
}

template IH53 {
    symbol [a+b+c+d+;b-a-c+d+]
    topology 4^4
    aspects 1,2,1r,2r
    colouring 2 (0 1 1 0) (0 1) (0 1)
    rules
        aspect 2 1:2
        aspect 1r 1:0
        aspect 2r 2:0
        translate T1 2:3
        translate T2 1:3,0,2,1
}

template IH54 {
    symbol [a+b+c+d+;a-b-c-d+]
    topology 4^4
    aspects 1,2,1r,2r
    colouring 2 (0 1 1 0) (0 1) (0 1)
    rules
        aspect 2 1:3
        aspect 1r 1:0
        aspect 2r 2:2
        translate T1 1r:2
        translate T2 2:1
}

template IH55 {
    symbol [a+b+c+d+;b+a+d+c+]
    topology 4^4
    aspects 1,2,3,4
    colouring 2 (0 1 0 1) (0 1) (0 1)
    rules
        aspect 2 1:1
        aspect 3 2:1
        aspect 4 3:1
        translate T1 2:2
        translate T2 4:3
}

template IH56 {
    symbol [a+b+c+d+;b+a+c-d-]
    topology 4^4
    aspects 1,2,3,4,1r,2r,3r,4r
    colouring 2 (0 1 0 1 1 0 1 0) (0 1) (0 1)
    rules
        aspect 2 1:1
        aspect 3 2:1
        aspect 4 3:1
        aspect 1r 2r:0
        aspect 2r 2:2
        aspect 3r 2r:1
        aspect 4r 3r:1
        translate T1 1r:3
        translate T2 2r:3,1
}

template IH57 {
    symbol [a+b+a+b+;a+b+]
    topology 4^4
    aspects 1
    colouring 2 (0) (1 0) (1 0)
    rules
        translate T1 1:0
        translate T2 1:1
}

template IH58 {
    symbol [a+b+a+b+;a-b+]
    topology 4^4
    aspects 1,1r
    colouring 2 (0 1) (1 0) (0 1)
    rules
        aspect 1r 1:0
        translate T1 1:1
        translate T2 1r:2
}

template IH59 {
    symbol [a+b+a+b+;b-a-]
    topology 4^4
    aspects 1,1r
    colouring 2 (0 1) (0 1) (0 1)
    rules
        aspect 1r 1:0
        translate T1 1r:2
        translate T2 1r:3
}

// IH60 can only be realized with marked templates.

template IH61 {
    symbol [a+b+a+b+;b+a+]
    topology 4^4
    aspects 1,2
    colouring 2 (0 1) (0 1) (0 1)
    rules
        aspect 2 1:0
        translate T1 2:0
        translate T2 2:2
}

template IH62 {
    symbol [a+a+a+a+;a+]
    topology 4^4
    aspects 1
    colouring 2 (0) (1 0) (1 0)
    rules
        translate T1 1:0
        translate T2 1:1
}

// IH63 can only be realized with marked templates.

template IH64 {
    symbol [a b+ c b-;c b- a]
    topology 4^4
    aspects 1
    colouring 2 (0) (1 0) (1 0)
    rules
        translate T1 1:0
        translate T2 1:1
}

// IH65 can only be realized with marked templates.

template IH66 {
    symbol [a b+ c b-;c b+ a]
    topology 4^4
    aspects 1,2
    colouring 2 (0 1) (1 0) (0 1)
    rules
        aspect 2 1:1
        translate T1 1:0
        translate T2 2:3
}

template IH67 {
    symbol [a b+ c b-;a b+ c]
    topology 4^4
    aspects 1,2
    colouring 2 (0 1) (0 1) (0 1)
    rules
        aspect 2 1:0
        translate T1 2:1
        translate T2 2:2
}

template IH68 {
    symbol [a+b+b-a-;b-a-]
    topology 4^4
    aspects 1
    colouring 2 (0) (1 0) (1 0)
    rules
        translate T1 1:0
        translate T2 1:1
}

template IH69 {
    symbol [a+b+b-a-;a+b+]
    topology 4^4
    aspects 1,2
    colouring 2 (0 1) (0 1) (0 1)
    rules
        aspect 2 1:0
        translate T1 2:1
        translate T2 2:2
}

// IH70 can only be realized with marked templates.

template IH71 {
    symbol [a+b+b-a-;b+a+]
    topology 4^4
    aspects 1,2,3,4
    colouring 2 (0 1 0 1) (0 1) (0 1)
    rules
        aspect 2 1:0
        aspect 3 2:0
        aspect 4 3:0
        translate T1 4:2
        translate T2 2:3
}

template IH72 {
    symbol [abab;ab]
    topology 4^4
    aspects 1
    colouring 2 (0) (1 0) (1 0)
    rules
        translate T1 1:0
        translate T2 1:1
}

template IH73 {
    symbol [a b a b;b a]
    topology 4^4
    aspects 1,2
    colouring 2 (0 1) (0 1) (0 1)
    rules
        aspect 2 1:0
        translate T1 2:0
        translate T2 2:2
}

template IH74 {
    symbol [a+a-a+a-;a+]
    topology 4^4
    aspects 1
    colouring 2 (0) (1 0) (1 0)
    rules
        translate T1 1:0
        translate T2 1:1
}

// IH75 can only be realized with marked templates.

template IH76 {
    symbol [a a a a;a]
    topology 4^4
    aspects 1
    colouring 2 (0) (1 0) (1 0)
    rules
        translate T1 1:0
        translate T2 1:1
}

template IH77 {
    symbol [a+b+c+;a-b-c-]
    topology 4.6.12
    aspects 1,2,3,4,5,6,1r,2r,3r,4r,5r,6r
    colouring 2 (0 0 0 0 0 0 1 1 1 1 1 1) (0 1) (0 1)
    rules
        aspect 2 1r:1
        aspect 3 2r:1
        aspect 4 3r:1
        aspect 5 4r:1
        aspect 6 5r:1

        aspect 1r 1:2
        aspect 2r 2:2
        aspect 3r 3:2
        aspect 4r 4:2
        aspect 5r 5:2
        aspect 6r 6:2

        translate T1 4r:0
        translate T2 3:0,1
}

template IH78 {
    symbol [a+b+c+;a-b+c-]
    topology 4.8^2
    aspects 1,2,1r,2r
    colouring 2 (0 1 1 0) (0 1) (1 0)
    rules
        aspect 2 1:1
        aspect 1r 2r:1
        aspect 2r 2:2
        translate T1 1r:2
        translate T2 2r:0
}

template IH79 {
    symbol [a+b+c+;c+b+a+]
    topology 4.8^2
    aspects 1,2,3,4
    colouring 2 (0 1 0 1) (1 0) (1 0)
    rules
        aspect 2 1:0
        aspect 3 2:0
        aspect 4 3:0
        translate T1 3:1
        translate T2 2:1,0
}

// IH80 can only be realized with marked templates.

template IH81 {
    symbol [a+b+c+;c+b-a+]
    topology 4.8^2
    aspects 1,2,3,4,1r,2r,3r,4r
    colouring 2 (0 1 0 1 1 0 1 0) (0 1) (0 1)
    rules
        aspect 2 1:0
        aspect 3 2:0
        aspect 4 3:0
        aspect 1r 4r:0
        aspect 2r 2:1
        aspect 3r 2r:0
        aspect 4r 3r:0
        translate T1 3r:1
        translate T2 4r:1,0
}

template IH82 {
    symbol [a+b a-;a-b]
    topology 4.8^2
    aspects 1,2,3,4
    colouring 2 (0 1 0 1) (1 0) (1 0)
    rules
        aspect 2 1:2
        aspect 3 2:2
        aspect 4 3:2
        translate T1 3:1
        translate T2 2:1,2
}

template IH83 {
    symbol [a+b+c+;b-a-c-]
    topology 6^3
    aspects 1,1r
    colouring 2 (0 1) (0 1) (0 1)
    rules 
        aspect 1r 1:2
        translate T1 1r:0
        translate T2 1r:1
}

template IH84 {
    symbol [a+b+c+;a+b+c+]
    topology 6^3
    aspects 1,2
    colouring 2 (0 1) (0 1) (0 1)
    rules
        aspect 2 1:0
        translate T1 2:1
        translate T2 2:2
}

template IH85 {
    symbol [a+b+c+;a-b+c+]
    topology 6^3
    aspects 1,2,1r,2r
    colouring 2 (0 1 1 0) (0 1) (0 1)
    rules
        aspect 2 1:1
        aspect 1r 2r:1
        aspect 2r 2:0
        translate T1 2:2
        translate T2 1r:0
}

template IH86 {
    symbol [a+b+c+;b-a-c+]
    topology 6^3
    aspects 1,2,1r,2r
    colouring 2 (0 1 1 0) (0 1) (0 1)
    rules
        aspect 2 2r:0
        aspect 1r 1:0
        aspect 2r 1r:2
        translate T1 1r:0
        translate T2 2:2
}

// IH87 can only be realized with marked templates.

template IH88 {
    symbol [a+b+c+;b+a+c+]
    topology 6^3
    aspects 1,2,3,4,5,6
    colouring 2 (0 1 0 1 0 1) (0 1) (0 1)
    rules
        aspect 2 1:0
        aspect 3 2:0
        aspect 4 3:0
        aspect 5 4:0
        aspect 6 5:0
        translate T1 4:2
        translate T2 5:2,1
}

// IH89 can only be realized with marked templates.

template IH90 {
    symbol [a+a+a+;a+]
    topology 6^3
    aspects 1,2
    colouring 2 (0 1) (0 1) (0 1)
    rules
        aspect 2 1:0
        translate T1 2:1
        translate T2 2:2
}

template IH91 {
    symbol [a b+b-;a b+]
    topology 6^3
    aspects 1,2
    colouring 2 (0 1) (0 1) (0 1)
    rules
        aspect 2 1:0
        translate T1 2:1
        translate T2 2:2
}

// IH92 can only be realized with marked templates.

template IH93 {
    symbol [a a a;a]
    topology 6^3
    aspects 1,2
    colouring 2 (0 1) (0 1) (0 1)
    rules
        aspect 2 1:0
        translate T1 2:1
        translate T2 2:2
}