Polyform Counts

Units are unit line segments on the triangular grid.

The polytrigs (fully-connected):

The quasi-polytrigs (includes disconnected forms that have gaps of maximum length 1):

"*" above means that pieces with enclosed holes exist.

Shapes

Holes (denoted by a "*" in the function name) consist of internal segments only, no circumference segments.

Triangles & holes:

T(n) = 3n(n + 1)/2

T*(n) = T(n) - 3n = T(n - 1)

e.g T(n=2) (n is the length of each side):

 /\ n=2
/__\

Hexagons & holes:

H(n) = 3n(3n + 1) = 6(T(n) - n)

H*(n) = H(n) - 6n = 3n(3n - 1)

e.g. H(2):

  ____
 /    \ n=2
/      \
\      /
 \____/

Hexagrams & holes:

Hg(n) = H(n) + 6T(n) - 6n
      = 6n(3n + 1)

Hg*(n) = Hg(n) - 12n
       = 6n(3n - 1)

e.g. Hg(n=2):

     /\ n=2
____/  \____
\          /
 \        /
 /        \
/___    ___\
    \  /
     \/

Elongated hexagons & holes:

He(m,n) = H(n) + (m - n)(6n + 1)
        = 3n² + 6mn + m + 2n

He*(m,n) = He(m,n) - 2m - 4n
         = 3n² + 6mn - m - 2n

When n==m:

He(n,n) == H(n)
He*(n,n) == H*(n)

e.g He(m=6,n=2) (m is the width of the bottom and n is the length of each of the near-vertical sides, not the overall height or width):

  ____________
 /            \ n=2
/              \
\              /
 \____________/
   m=6

Semi-regular hexagons (two different side lengths, alternating; == truncated triangles) & holes (m > n):

Hs(m,n) = T(m + 2n) - 3T(n) + 3n

Hs*(m,n) = Hs(m,n) - 3(m + n)

e.g. Hs(m=2,n=1):

  __
 /  \ m=2
/    \
\____/ n=1

Parallelograms & holes:

P(m,n) = 3mn + m + n

P*(m,n) = 3mn - (m + n)

e.g. P(m=4,n=2):

  ________
 /       /
/_______/ n=2
  m=4

Trapezoids & holes:

Tr(m,n) = (6mn - 3n² + 2m + n)/2

Tr*(m,n) = Tr(m,n) - 2m - n

e.g. T(m=4,n=2):

  ____
 /    \ n=2
/______\
  m=4

Chevrons & holes:

C(m,n) = 6mn + m + 2n

C*(m,n) = 6mn - (m + 2n)

Chevrons == parallelograms:

C(m,n) == P(m,2n)

e.g. C(m=3,n=1):

______
\     \ n=1
/_____/
  m=3

Butterflies & holes:

B(m,n) = 6mn - 3n² + m + 2n

B*(m,n) = B(m,n) - 2m - 4n
        = 6mn - 3n² - (m + 2n)

e.g. B(m=3,n=1) (m is the length of the base, n is the length of each tilted side):

______
\    / n=1
/____\
  m=3

Potential Puzzles

Puzzles not otherwise noted below have not been implemented or solved.

Initial numbers are the counts of unit line segments in the puzzles.

7: polytrigs O(1) + O(2)

18: quasi-ditrigs

• Potential:
  • He(2,1) - 1 unit
  • 2P(2,1) (= Tr(5,1) - 1 unit)
  • B(3,1) - 2 units
• No solutions:
  • Hg(1) - 6 units (= QuasiDitrigsHexagram)

19: quasi-polytrigs O(1) + O(2)

• Tr(4,2) - 4 units
• C(3,1) - 4 units
• B(3,1) - 1 unit

24: snake tritrigs

27: unwelded tritrigs

36: tritrigs

• Potential:
  • T(3) + 3Tr(2,1) spinner
  • H(1) + 6P(1,1) spinner
• No solutions:
  • H(2) - H*(1) (= TritrigsHex2Ring)
  • Hs(3,1) - 3 segments (= TritrigsTrefoil1, TritrigsTrefoil2, TritrigsWhorl)
  • T(4) + 3T(1) (= TritrigsSpikedTriangle2)
  • T(5) - various holes (= TritrigsTriangle_x)
  • OneSidedTritrigsTrilobedCuboid_x

42: ditrigs & tritrigs

• Potential:
  • more solutions to 3 congruent groups?
• No solutions:
  • many attempts to solve 3 congruent groups (= Polytrigs23ThreeCongruent_x1, ...)
  • Polytrigs23TriangleStack_x

42: one-sided snake tritrigs

43: polytrigs O(1) + O(2) + O(3)

45: one-sided unwelded tritrigs

49: one-sided snake polytrigs O(1) + O(2) + O(3)

52: one-sided unwelded polytrigs O(1) + O(2) + O(3)

57: one-sided tritrigs

• No solutions:
  • C(1,7)
  • P(8,2) - 1 segment
  • P(6,3) - H*(1)

64: one-sided polytrigs O(1) + O(2) + O(3)

• P(5,4) - P*(3,1)

128: snake tetratrigs

133: unwelded tetratrigs, including 1 extra segment for hole in O4 (said 1 segment hole must be removed from coordinates)

159: snake polytrigs O(1) + O(2) + O(3) + O(4)

166: unwelded polytrigs O(1) + O(2) + O(3) + O(4) (hole in O4 filled by I1)

241: 240+1: tetratrigs, including 1 extra segment for hole in O4 (said 1 segment hole must be removed from coordinates)

• He(11,3) - 1 seg (* = 241, -1 more seg for hole)
• T(13) - H*(2) - two more segments
• P(12,7) - H*(2)
• Tr(13,12) - H*(2)
• ditto for T*(5) instead of H*(2)
• Tr(14,13) - H*(3)
• P(13,6) - Hg*(1)
• Tr(14,13) - Hg*(1)
• Tr(13,9) - H*(1)
• P(11,7) - P*(2,2)
• P(15,5) - Tr*(3,1)

265: one-sided snake polytrigs O(1) + O(2) + O(3) + O(4)

272: one-sided unwelded polytrigs O(1) + O(2) + O(3) + O(4) (hole in O4 filled by I1)

283: polytrigs O(1) + O(2) + O(3) + O(4) (hole in O4 filled by I1)

• Tr(15,8)
• P(11,8)
• Hg(4) - Tr*(5,4)
• Hg(4) - P*(4,2)
• B(16,3)
• C(3,14)
• C(11,4)

417: one-sided tetratrigs, including 1 extra segment for hole in O4

• He(6,1) - Tr*(2,1)
• P(12,11) - Tr*(2,1)
• C(11,6) - Tr*(2,1)
• B(14,6) - B*(2,1)
• He(11,5) - T*(3)
• He(3,9) - T*(3)
• B(13,7) - T*(3)
• P*(15,9) - 2 x H*(1)
• P*(15,9) - 2 x Tr*(2,1)
• P*(15,9) - Hg*(1)
• P*(15,9) - Tr*(7,1)
• Hs*(8,6)

420: quasi-tritrigs

439: quasi-polytrigs O(1) + O(2) + O(3)

480: one-sided polytrigs O(1) + O(2) + O(3) + O(4) (hole in O4 filled by I1)

• Hg(5)
• 2 x H(5)

Misc

• Idea from Leslie E. Shader: 8 polytrigs on a tetrahedron, no bending allowed (polytrigs remain planar). (Video in 2012-01-23 email.)

Links