A110312 -id:A110312 - cp's OEIS frontend

A362938 a(n) = minimum number of pieces needed to dissect a regular n-sided polygon into a monotile, i.e. a polygonal tile which tiles the plane (conjectured).

1, 1, 2, 1, 3, 2, 3, 2, 4, 3, 4, 3, 5, 4, 6

Offset: 3

N. J. A. Sloane, Aug 29 2023

I do not know which of these values have been proved to be minimal. Probably only a(n) for n = 3, 4, 5, 6, 8, and 10.
The three related sequences A110312, A362938, A362939 are exceptions to the usual OEIS policy of requiring that all terms in sequences must be known exactly. These sequences are included because of their importance and in the hope that someone will establish the truth of some of the conjectured values.
More formally, a(n) = minimum number of pieces needed to dissect a regular n-sided polygon into a monotile, that is, a prototile for a monohedral tiling of the plane.
Turning over is allowed. The pieces must be bounded by simple curves to avoid difficulties with non-measurable sets.
The definitions imply that a(n) <= A362939(n) <= A110312(n).
On Aug 31 2023 Gavin Theobald sent me two different solutions for a(9) = 3 and one solution for a(11) = 4 (see links). He reports that he found these dissections in the 1990's. In his email and in a later email (Sep 04 2023) he also gives the values a(13) = 5, a(14) = 3, a(15) = 5 (with one piece turned over), a(16) = a(18) = 4, a(20) = 5. He conjectures that a(2t) = floor(t/2) for all t >= 2.

Branko Grünbaum and G. C. Shephard, Tilings and Patterns. W. H. Freeman, New York, 1987. [The sequence is defined in Section 2.6, pp. 91-95.]
Harry Lindgren, Geometric Dissections, Van Nostrand, Princeton, NJ, 1964. Plates B6, B7, B8, B9, B10, and B12 illustrate n = 6, 7, 8, 9, 10, and 12, respectively. One would expect that plates B11 and B13 would refer to n = 11 and 13, but in fact they appear to give alternative solutions for n = 10 and 12, respectively.
Harry Lindgren, Recreational Problems in Geometric Dissections and How to Solve Them, Revised and enlarged by Greg Frederickson, Dover Publications, NY, 1972.

Branko Grünbaum and G. C. Shephard, Tilings and Patterns, W. H. Freeman, New York, 1987. Annotated copy of Fig. 2.6.1, illustrating a(5), a(7), a(8), a(10), and a(12). (Their value for a(9) is out-of-date.)
Harry Lindgren, Geometric Dissections, Annotated scan of Plate B12, showing tiling of plane arising from the conjectured a(12) = 3.
N. J. A. Sloane, Illustration for a(5) = 2, after Grunbaum and Shephard, Fig. 2.6.1. Left: The 2-piece dissection of the pentagon. Right: Shows how the hexagonal tile made from those two pieces tiles the plane.
N. J. A. Sloane, An illustration for a(12) = 3, based on Lindgren's plate B12.
N. J. A. Sloane, New Gilbreath Conjectures, Sum and Erase, Dissecting Polygons, and Other New Sequences, Doron Zeilberger's Exper. Math. Seminar, Rutgers, Sep 14 2023: Video, Slides, Updates. (Mentions this sequence.)
N. J. A. Sloane and Gavin A. Theobald, On Dissecting Polygons into Rectangles, arXiv:2309.14866 [math.CO], 2023.
Gavin Theobald, Illustration for a(5)
Gavin Theobald, Illustration for a(7)
Gavin Theobald, Another illustration for a(7) <= 3, after Lindgren.
Gavin Theobald, Illustration for a(8)
Gavin Theobald, A 9-piece dissection of a 9-gon into a monotile, with the monotile outlined in red, illustrating a(9) = 3.
Gavin Theobald, A 9-piece dissection of a 9-gon into a monotile, showing how the monotile is obtained from the 9-gon.
Gavin Theobald, An alternative illustration for a(9) = 3.
Gavin Theobald, Yet another illustration for a(9)
Gavin Theobald, Illustration for a(10)
Gavin Theobald, Illustration for a(11)
Gavin Theobald, Illustration for a(12)
Gavin Theobald, Illustration for a(13) <= 4
Gavin Theobald, Illustration for a(14)
Gavin Theobald, Illustration for a(15) (5 pieces)
Gavin Theobald, Another illustration for a(15) <= 5
Gavin Theobald, Illustration for a(17) (The piece marked X must be turned over)
Gavin Theobald, Illustration for a(19) <= 7
Gavin Theobald, The Geometric Dissections Database

Cf. A110312, A362939.

a(9) = 3, a(11) = 4, a(13) = 5, a(14) = 3, a(16) = 4 from Gavin Theobald, Aug 31 2023 - Sep 11 2023.

Updated with many further illustrations from Gavin Theobald. - N. J. A. Sloane, Sep 19 2023

A362939 a(n) = minimum number of pieces needed to dissect a regular n-sided polygon into a rectangle (conjectured).

Original entry on oeis.org

2, 1, 4, 3, 5, 4, 7, 4, 9, 5, 10, 7, 10, 9

Offset: 3

N. J. A. Sloane, Aug 31 2023

The dimensions of the rectangle can be anything you want, as long as it is a rectangle.
Turning over is allowed. The pieces must be bounded by simple curves to avoid difficulties with non-measurable sets.
Apart from changing "square" to "rectangle", the rules are the same as in A110312.
I do not know which of these values have been proved to be minimal. Probably only a(3)=2 and a(4)=1.
The three related sequences A110312, A362938, A362939 are exceptions to the usual OEIS policy of requiring that all terms in sequences must be known exactly. These sequences are included because of their importance and in the hope that someone will establish the truth of some of the conjectured values.
The definitions imply that A362938(n) <= a(n) <= A110312(n).

			See our paper "On dissecting polygons into rectangles" for illustrations of a(n) for all n <= 16 except n=13 and n=15.

Adam Gsellman, Illustration for a(5) <= 4, a 4-piece dissection of a regular pentagon to a rectangle, May 16 2023.
Adam Gsellman, Another construction showing that a(5) <= 4, May 16 2023.
Adam Gsellman, Illustration for r(8) <= 4, a 4-piece dissection of a regular octagon to a rectangle, May 16 2023.
Adam Gsellman, First 4-piece dissection of a regular octagon to a rectangle, showing details of the dissection [Needs a very wide window to see full illustration]
Adam Gsellman, Another construction showing that a(8) <= 4, May 16 2023.
N. J. A. Sloane, Another 4-piece dissection of a regular pentagon to a rectangle, showing a(5) <= 4, Jun 08 2023.
N. J. A. Sloane, Illustrating a(6) <= 3: three-piece dissection of regular hexagon to a rectangle. (Surely there is a proof that this cannot be done with only two pieces?)
N. J. A. Sloane, Illustration 12gonA for a(12) <= 5, a 5-piece dissection of a regular dodecagon to a rectangle, May 18 2023.
N. J. A. Sloane, Illustration 12gonB2 for a(12) <= 5, showing the rearranged pieces.
N. J. A. Sloane, Illustration 12gonC for a(12) <= 5, showing vertex and edge labels.
N. J. A. Sloane, Illustration 12gonD for a(12) <= 5, giving proof of correctness.
N. J. A. Sloane, New Gilbreath Conjectures, Sum and Erase, Dissecting Polygons, and Other New Sequences, Doron Zeilberger's Exper. Math. Seminar, Rutgers, Sep 14 2023: Video, Slides, Updates. (Mentions this sequence.)
N. J. A. Sloane and Gavin A. Theobald, On Dissecting Polygons into Rectangles, arXiv:2309.14866 [math.CO], 2023.
Gavin Theobald, A 7-piece dissection of a 9-gon to a rectangle (See our paper "On dissecting polygons into rectangles" for details of this dissection)
Gavin Theobald, A 4-piece dissection of a 10-gon to a rectangle (See our paper "On dissecting polygons into rectangles" for details of this dissection)
Gavin Theobald, The Geometric Dissections Database

Cf. A110312, A362938.

A110000 Minimal number of polygonal pieces in a dissection of a regular n-gon to an equilateral triangle (conjectured).

Original entry on oeis.org

1, 4, 6, 5, 8, 7, 8, 7

Offset: 3

N. J. A. Sloane, Sep 11 2005

nonn

I do not know which of these values have been proved to be minimal.

Turning over is allowed. The pieces must be bounded by simple curves to avoid difficulties with non-measurable sets.

			a(3) = 1 trivially.
a(4) <= 4 because there is a 4-piece dissection of an equilateral triangle into a square, due probably to H. Dudeney, 1902 (or possible C. W. McElroy - see Fredricksen, 1997, pp. 136-137). Surely it is known that this is minimal? See illustrations.
Coffin gives a nice description of this dissection. He notes that the points marked * are the midpoints of their respective edges and that ABC is an equilateral triangle. Suppose the square has side 1, so the triangle has side 2/3^(1/4). Locate B on the square by measuring 1/3^(1/4) from A, after which the rest is obvious.
For n >= 5 see the Theobald web site.

G. N. Frederickson, Dissections: Plane and Fancy, Cambridge, 1997.
H. Lindgren, Geometric Dissections, Van Nostrand, Princeton, 1964.
H. Lindgren (revised by G. N. Frederickson), Recreational Problems in Geometric Dissections and How to Solve Them, Dover, NY, 1972.

Stewart T. Coffin, Dudeney's 1902 4-piece dissection of a triangle to a square, from The Puzzling World of Polyhedral Dissections.
Stewart T. Coffin, The Puzzling World of Polyhedral Dissections, Chapter 1. (See section "Geometrical Dissections".)
Geometry Junkyard, Dissection
Gavin Theobald, Triangle dissections
Vinay Vaishampayan, Dudeney's 1902 4-piece dissection of a triangle to a square

Cf. A110312, A110356.

A110356 Array read by antidiagonals: T(n,k) (n>=3, k>=3) = minimal number of polygonal pieces in a dissection of a regular n-gon to a regular k-gon (conjectured).

Original entry on oeis.org

1, 4, 4, 6, 1, 6, 5, 6, 6, 5

Offset: 3

N. J. A. Sloane, Sep 11 2005

I do not know which of these values have been proved to be minimal.
Turning over is allowed. The pieces must be bounded by simple curves to avoid difficulties with non-measurable sets.
See A110312 for further references and illustrations.

			Array begins:
1 4 6 5 8 7 ... <= A110000
4 1 6 5 7 5 9 ... <= A110312
6 6 1 7 ...
5 5 7 1 ...
8 7 ...
...

G. N. Frederickson, Dissections: Plane and Fancy, Cambridge, 1997.

Row 3 gives A110000, row 4 gives A110312.

A214367 Denominators of continued fraction convergents to ( sqrt(4*sqrt(3)-3) - 1 )/4.

Original entry on oeis.org

1, 4, 53, 57, 110, 167, 277, 444, 721, 119409, 120130, 239539, 1557364, 1796903, 3354267, 15213971, 109852064, 2871367635, 2981219699, 14796246431, 17777466130, 50351178691, 68128644821, 186608468333, 254737113154, 696082694641, 950819807795, 2597722310231, 13939431358950

Offset: 1

Jean-Claude Babois, Feb 20 2013

This constant arises as the edge-length of one of the pieces in the classical 4-piece dissection of a square to an equilateral triangle (see references in A110312).

Cf. A224224, A110312.

Denominator[Convergents[(Sqrt[4 Sqrt[3] - 3] - 1)/4, 50]]

Edited by N. J. A. Sloane, Apr 08 2013 at the request of Jean-Claude Babois

Mathematica code corrected to agree with terms by Ray Chandler, Mar 13 2017

A141292 Conjectured values for minimal number of pieces required in a 2n-gon to square dissection that uses translation alone.

Original entry on oeis.org

1, 5, 9, 12, 15, 19, 22, 25, 28, 31, 35, 38, 42, 47, 50, 53, 56, 60, 63, 67, 72, 76, 79, 82, 85, 89, 93, 100, 103, 106, 109, 113, 117, 121, 126, 130, 133, 136, 139, 143, 147, 156, 160, 163, 166, 169, 174, 177, 182, 186, 189, 192, 196, 202, 205, 214, 217, 220, 223

Offset: 2

Pamela Pierce (PPierce(AT)wooster.edu), Jeffrey Willert (jwillert09(AT)wooster.edu) and Wenyuan Wu (wwu11(AT)wooster.edu), Aug 01 2008, Aug 12 2008

nonn

			a(2)=1 because a regular 4-gon-to-square dissection can be accomplished with a single "piece". Busschop gave a 5-piece hexagon-to-square dissection using translations alone,so a(3)=5 (see Frederickson, p. 118). Further terms in the sequence are obtained by a systematic process for cutting the original 2n-gon, and the algorithm for generating these terms is given below. - Pamela Pierce (ppierce(AT)wooster.edu), Sep 03 2009

Boltyanskii, V.G., Equivalent and Equidecomposable Figures, D.C. Heath and Company, Boston, 1963.
G. N. Frederickson, Dissections Plane and Fancy, Camb. 1997.

Cf. A110312.

b1 := (n, k) -> 2*sin(Pi*(2*k-1)/n)
b2 := (n, k) -> 2*sin(Pi*(2*k+1)/n)
w1 := (n, k) -> b1(n, k)+b2(n, k)
w2 := (n, k) -> sqrt((1/2)*n*sin(2*Pi/n))
h1 := (n, k) -> cos(Pi*(2*k-1)/n)-cos(Pi*(2*k+1)/n)
h2 := (n, k) -> w1(n, k)*h1(n, k)/w2(n, k)
a := (n, k) -> floor(w2(n, k)/w1(n, k))*h2(n, k)/h1(n, k)
kp := (n, k) -> 3*signum(w1(n, k)-w2(n, k))+3+((1/2)*signum(w2(n, k)-w1(n, k))+1/2)* (3*floor(w2(n, k)/w1(n, k))+9/2+(1/2)*signum(w2(n, k)-floor(w2(n, k)/w1(n, k))*w1(n, k)-(1/2-(1/2)*a(n, k))*w1(n, k)-a(n, k)*b2(n, k)))
P := n-> 9/2+sum(kp(n, k), k = 1 .. floor((1/4)*n)-1)+1.5*signum((1/4)*n-floor((1/4)*n)-.25)
[seq([2*i, P(2*i)], i = 3 .. 100)]

Entry revised by Pamela Pierce (PPierce(AT)wooster.edu), John Ramsay (JRamsay(AT)wooster.edu), Jeffrey Willert (jawiller(AT)ncsu.edu), Hannah Roberts (HRoberts12(AT)wooster.edu), Nancy Tinoza (NTinoza12(AT)wooster.edu) and Wenyuan Wu (wwu11(AT)wooster.edu), Sep 03 2009. The revisions are based on a dissection method found in 2009.

Given that the value of a(3) changed from 6 to 5 at the latest revision, one should not have too much confidence that these entries are minimal. - N. J. A. Sloane, Sep 05 2009

cp's OEIS Frontend

A362938 a(n) = minimum number of pieces needed to dissect a regular n-sided polygon into a monotile, i.e. a polygonal tile which tiles the plane (conjectured).

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A362939 a(n) = minimum number of pieces needed to dissect a regular n-sided polygon into a rectangle (conjectured).

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A110000 Minimal number of polygonal pieces in a dissection of a regular n-gon to an equilateral triangle (conjectured).

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A110356 Array read by antidiagonals: T(n,k) (n>=3, k>=3) = minimal number of polygonal pieces in a dissection of a regular n-gon to a regular k-gon (conjectured).

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A214367 Denominators of continued fraction convergents to ( sqrt(4*sqrt(3)-3) - 1 )/4.

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A141292 Conjectured values for minimal number of pieces required in a 2n-gon to square dissection that uses translation alone.

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