Pentagon tiling (type 1 & 2)

After reading an interesting article on pentagon tilings by David Bailey, it reminded me of a shape I had constructed some years ago, from four isosceles triangles with sides 1,2 and 2.  Jaap Scherphuis commented that the tile is not only of type 1 (from A+B=180) but also of type 2 (not only from B+D=180 with b=d, but also from B+D=180 with c=e). 


B+A=180, B+D=180, B+E=180, C=2B, a=b=d, c=e, a=2c

Below are a few periodic tilings that be constructed with this shape.  It can also tile chaotically although I haven’t got too far with it yet.  Coloured areas denote mirroring.


2 thoughts on “Pentagon tiling (type 1 & 2)

  1. Dear Mr. Smith!
    (Jaap Scherphuis references your blog as “Dave Smith”.)

    1) Could You give detailed references to the article by David Bailey You read?
    2) Would that be David H. Bailey, the co-developer of the PLSQ
    (integer relation) algorithm and the Bailey-Borwein-Plouffe-formula
    (allowing the n-th hexadecimal digit of \pi to be computed WITHOUT
    previous computation of the previous digits)?
    3) I name the current pentagon P4 (dissectable into either 4 isosceles
    1-2-2 triangles or a wide and a narrow kite — together into 4 triangles of
    [necessarily] same area, which You show in later blogs).
    I object to the statement “but also of type 2
    (not only from B+D=180 with b=d, but also from B+D=180 with c=e)”, because
    this insunuates that the conditions on the lengths would be independant of
    each other, that one could have one without the other (which one can’t, at
    least not with the standard tilings according to type).
    A good definition of a type 2 pentagon would be, that it admits a labeling
    of consecutive corners with B+D=180° AND (b=d OR c=e) and if and only if
    BOTH conditions on the lengths are true together, then this pentagon admits
    an edge-to-edge (EE) periodic tiling of the euclidean plane (as You show).
    When one fixes B (and e) in Your labeling of P4, but instead continues
    clockwise one finds
    3.1) B+C=180° (type 1), but NEITHER of the EE-types 1
    B+C=180°, b=d (EE-type 1a),
    B+C=180°, A=D, a=e (EE-type 1b),
    thus every tiling as type 1 is NOT EE by necessity (as You show), and
    3.2) B+D=180° (probable type 2), but NEITHER b=d NOR c=e,
    thus no type 2 after all (which I find remarkable).
    Together I would state, that P4 can be seen (in essentially ONE way) as
    being of EE-type 2 an (in essentially one way) as (not-EE) type 1.
    4) Similarily with P3, dissectable into a narrow kite and an 1-2-2 triangle
    (Your blog dated 29. Aug 2016).
    I would state, that P3 can be seen in TWO ways as type 2:
    either as EE-type 2 (which You show there, but that tiling uses 2 of Your
    ways, not just one)
    or as a not-EE type 2 (which you show there, using Your third way).

    Thanks a lot for bearing with me,
    Bernhard W. Marx,
    Dresden, Germany,


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