Editing Polydrafter

{{Short description|Geometric shape formed of right triangles}}
[[File:Monodrafter.png|thumb|30–60–90 triangle]]
In [[recreational mathematics]], a '''polydrafter''' is a [[polyform]] with a [[Special right triangles#30–60–90 triangle|30°–60°–90°]] [[right triangle]] as the base form. This triangle is also called a [[set square|drafting triangle]], hence the name.<ref>{{citation
 | last1 = Salvi | first1 = Anelize Zomkowski
 | last2 = Simoni | first2 = Roberto
 | last3 = Martins | first3 = Daniel
 | editor1-last = Dai | editor1-first = Jian S.
 | editor2-last = Zoppi | editor2-first = Matteo
 | editor3-last = Kong | editor3-first = Xianwen
 | contribution = Enumeration problems: A bridge between planar metamorphic robots in engineering and polyforms in mathematics
 | doi = 10.1007/978-1-4471-4141-9_3
 | pages = 25–34
 | publisher = Springer
 | title = Advances in Reconfigurable Mechanisms and Robots I
 | year = 2012| isbn = 978-1-4471-4140-2
 }}.</ref> This triangle is also half of an [[equilateral triangle]], and a polydrafter's cells must consist of halves of triangles in the [[triangular tiling]] of the plane; consequently, when two drafters share an edge that is the middle of their three edge lengths, they must be reflections rather than rotations of each other. Any contiguous subset of halves of triangles in this tiling is allowed, so unlike most polyforms, a polydrafter may have cells joined along unequal edges: a hypotenuse and a short leg.

==History==
Polydrafters were invented by [[Christopher Monckton, 3rd Viscount Monckton of Brenchley|Christopher Monckton]], who used the name ''polydudes'' for polydrafters that have no cells attached only by the length of a short leg.  Monckton's [[Eternity Puzzle]] was composed of 209 12-dudes.<ref>{{citation|title=The Math Book: From Pythagoras to the 57th Dimension, 250 Milestones in the History of Mathematics|first=Clifford A.|last=Pickover|authorlink=Clifford A. Pickover|publisher=Sterling Publishing Company, Inc.|year=2009|isbn=9781402757969|page=496|url=https://books.google.com/books?id=JrslMKTgSZwC&pg=PA496}}.</ref>

The term ''polydrafter'' was coined by [[Ed Pegg Jr.]], who also proposed as a puzzle the task of fitting the 14 tridrafters&mdash;all possible clusters of three drafters&mdash;into a trapezoid whose sides are 2, 3, 5, and 3 times the length of the hypotenuse of a drafter.<ref>{{citation|contribution=Polyform Patterns|first=Ed Jr.|last=Pegg|authorlink=Ed Pegg Jr.|pages=119–125|title=Tribute to a Mathemagician|editor1-first=Barry|editor1-last=Cipra|editor1-link=Barry Arthur Cipra|editor2-first=Erik D.|editor2-last=Demaine|editor2-link=Erik Demaine|editor3-first=Martin L.|editor3-last=Demaine|editor3-link=Martin Demaine|editor4-first=Tom|display-editors = 3 |editor4-last=Rodgers|publisher=A K Peters|year=2005}}.</ref>

==Extended polydrafters==

[[File:Extended_didrafters.png|thumb|Two extended didrafters]]
An ''extended polydrafter'' is a variant in which the drafter cells cannot all conform to the triangle ([[polyiamond]]) grid.
The cells are still joined at short legs, long legs, hypotenuses and half-hypotenuses.
See the Logelium link below.

==Enumerating polydrafters==

Like [[polyomino]]es, polydrafters can be enumerated in two ways, depending on whether [[Chirality (mathematics)|chiral]] pairs of polydrafters are counted as one polydrafter or two.

{| class="wikitable" align="center" cellpadding="4" cellspacing="0"
! rowspan="2" | ''n''
! rowspan="2" | Name of<br/>''n''-polydrafter
! colspan="2" | Number of ''n''-polydrafters<br><div style="font-size:83%; line-height:120%">(reflections counted separately)</div>
! rowspan="2" | Number<br>of free<br>''n''-polydudes
|-
! style="max-width:0;" | free      <br/><div style="font-size:58%; line-height:120%">{{OEIS|id=A056842}}</div>
! style="max-width:0;" | one-sided <br/><div style="font-size:58%; line-height:120%">{{OEIS|id=A217720}}</div>
|-
|1
|monodrafter
|align=right|1
|align=right|2
|align=right|1
|-
|2
|didrafter
|align=right|6
|align=right|8
|align=right|3
|-
|3
|tridrafter
|align=right|14
|align=right|28
|align=right|1
|-
|4
|tetradrafter
|align=right|64
|align=right|116
|align=right|9
|-
|5
|pentadrafter
|align=right|237
|align=right|474
|align=right|15
|-
|6
|hexadrafter
|align=right|1024
|align=right|2001
|align=right|59
|}

With two or more cells, the numbers are greater if extended polydrafters are included.  For example, the number of didrafters rises from 6 to 13.  See {{OEIS|id=A289137}}.

==See also==
*The [[kisrhombille tiling]], a tessellation of the plane made of 30°–60°–90° triangles.

==References==
{{Reflist}}

==External links==
*{{MathWorld|urlname=Polydrafter|title=Polydrafter}}
*[http://www.mathpuzzle.com/eternity.html The Eternity Puzzle], at [http://www.mathpuzzle.com/ mathpuzzle.com]
*[http://www.logelium.de/Drafter/PolyDrafter_EN.htm Drafter], at [http://www.logelium.de/ Logelium]

{{Polyforms}}

[[Category:Polyforms]]