17:
197:; it takes the form of an irregular equidiagonal pentagon with an obtuse isosceles triangle attached to one of its sides, with the distance from the apex of the triangle to the opposite pentagon vertex equal to the diagonals of the pentagon. Its area is 0.674981.... (sequence
156:
with unit-length diagonals, which has area 1/2. However, infinitely many other orthodiagonal and equidiagonal quadrilaterals also have diameter 1 and have the same area as the square, so in this case the solution is not unique.
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must themselves be at most 1. Therefore, the quadrilateral has largest area when the three factors in the area formula are individually maximized, with
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275: = 8 this was verified by a computer calculation by Audet et al. Graham's proof that his hexagon is optimal, and the computer proof of the
311:, the polygons maximizing perimeter for their diameter, maximizing width for their diameter, and maximizing width for their perimeter
267: − 1)-gon with an isosceles triangle attached to one of its sides, its apex at unit distance from the opposite (
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The full conjecture of Graham, characterizing the solution to the biggest little polygon problem for all even values of
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Biggest little polygon with 6 sides (on the left); on the right the regular polygon with same diameter but lower area.
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Foster, Jim; Szabo, Tamas (2007), "Diameter graphs of polygons and the proof of a conjecture of Graham",
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Audet, Charles; Hansen, Pierre; Messine, Frédéric; Xiong, Junjie (2002), "The largest small octagon",
207:), a number that satisfies the equation (although not expressible in radicals due to it having the
189:= 6, the unique optimal polygon is not regular. The solution to this case was published in 1975 by
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is either of the angles they form with each other. In order for the diameter to be at most 1, both
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are within unit distance of each other) and that has the largest
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144:(its diagonals have equal length), and the condition that sin(
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is an odd number, but the solution is irregular otherwise.
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Schäffer, J. J. (1958), "Nachtrag zu Ungelöste Prob. 12",
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Jahresbericht der
Deutschen Mathematiker-Vereinigung
177:has largest area among all diameter-one polygons.
108:are the two diagonals of the quadrilateral and
271: − 1)-gon vertex. In the case
263:consists in the same way of an equidiagonal (
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358:"Extremale Polygone gegebenen Durchmessers"
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294:, was proven in 2007 by Foster and Szabo.
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490:Journal of Combinatorial Theory
451:Journal of Combinatorial Theory
420:Journal of Combinatorial Theory
84:= 4, the area of an arbitrary
44:one (that is, every two of its
549:Graham's Largest Small Hexagon
383:(1956), "Ungelöste Prob. 12",
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434:10.1016/0097-3165(75)90004-7
255: + 11993 = 0.
551:, from the Hall of Hexagons
412:"The largest small hexagon"
150:orthodiagonal quadrilateral
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504:10.1016/j.jcta.2007.02.006
142:equidiagonal quadrilateral
239: − 221360
536:"Biggest Little Polygon"
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88:is given by the formula
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52:among all diameter-one
465:10.1006/jcta.2001.3225
304:Hansen's small octagon
26:biggest little polygon
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287:with straight edges.
181:Even numbers of sides
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161:Odd numbers of sides
247: + 144464
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532:Weisstein, Eric W.
385:EIemente der Math.
331:Elemente der Math.
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559:Categories
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100:)/2 where
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541:MathWorld
368:: 251–270
285:thrackles
60:= 4 is a
40:that has
410:(1975),
381:Lenz, H.
356:(1922),
298:See also
283:-vertex
42:diameter
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337:: 85–86
203:in the
200:A111969
38:polygon
36:-sided
32:is the
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154:square
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415:(PDF)
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570:Area
391:: 86
205:OEIS
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