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Staff Picks Mathematics Equation Solving Geometry High-Performance Computing Wolfram Language Optimization

A closed form expression for the area of the 'biggest little’ octagon and decagon

Greg Hurst

Posted 5 days ago

The 'largest small octagon' is the 8-sided polygon of unit diameter having largest possible area. In 1975, Ronald Graham found a closed form expression for the area of the analogous hexagon -- a root of an irreducible degree 10 polynomial (found here). In 2002, Audet et al. found the value of maximal octagon area to be ~0.726867 and as far as I could find, no closed form expression has been found. Below is a picture of the maximizing hexagon and octagon (found here): Here, the red lines indicate the vertices a unit distance from each other. If we make some assumptions on symmetry, we can have simplified labels of the octagon's vertices and come up with constraints on them: Our constraints will Place bounds on location: 0 < x1 < 1/2 && 0 < x3 < 1/2 && 0 < y1 < 1/2 && 0 < y2 < 1/2 && 1/2 < y3 < 1 Add unit diameter constraints: (x1 + x3)^2 + (y1 - y3)^2 == 1 && x1^2 + (y1 - 1)^2 == 1 && (1/2 + x3)^2 + (y2 - y3)^2 == 1 We aim to maximize area, which is given by: x3 - y1/2 + x1 y2 - x3 y2 + y3/2 Unfortunately a (symbolic) call to `Maximize` did not return a result in a reasonable amount of time: area = x3 - y1/2 + x1 y2 - x3 y2 + y3/2; bounds = {0 < x1 < 1/2, 0 < x3 < 1/2, 0 < y1 < 1/2, 0 < y2 < 1/2, 1/2 < y3 < 1}; cons = {(x1 + x3)^2 + (y1 - y3)^2 == 1, x1^2 + (y1 - 1)^2 == 1, (1/2 + x3)^2 + (y2 - y3)^2 == 1}; Maximize[{area, Join[bounds, cons]}, {x1, x3, y1, y2, y3}] (* $Aborted ) Numerical optimization works nonetheless but slows down quickly as working precision goes up: NMaximize[{area, Join[bounds, cons]}, {x1, x3, y1, y2, y3}, WorkingPrecision -> 1000, MaxIterations -> 1000]; // AbsoluteTiming ( {675.194, Null} ) To get higher precision results faster, we can do some symbolic preprocessing to reduce our parameter space: cons2 = Reduce[And @@ Join[bounds, cons], {x1, y1, x3}, Reals, Backsubstitution -> True][[1]]; area2 = area /. ToRules[cons2 /. _Inequality -> True]; res10000 = NMaximize[{area2, cons2 /. _Equal -> True}, {y2, y3}, WorkingPrecision -> 10000]; // AbsoluteTiming ( {7.24073, Null} ) Notice the ~100x speedup despite increasing the working precision itself 10x. Since the call to `Maximize` didn't work, we can take matters into our own hands: Formulate the problem through Lagrangian multipliers: consdiff = Subtract @@@ cons; L = area - {λ1, λ2, λ3}.consdiff; ds = D[L, #] & /@ {x1, x3, y1, y2, y3}; cons3 = Join[ds, consdiff, {area - m}]; Use `GroebnerBasis` to find a polynomial relation (the magic step): gb = GroebnerBasis[cons3, {m}, {x1, x3, y1, y2, y3, λ1, λ2, λ3}]; MatchQ[gb, {poly_} /; PolynomialExpressionQ[poly, m, NumericQ]] ( True ) Select the valid root as the area maximizer, and verify it symbolically: Length[sols = Solve[gb[[1]] == 0, m, PositiveReals]] ( 9 ) solQ = ParallelMap[ Solve[ Join[Thread[cons3 == 0], bounds, {m == (m /. #)}], {x1, x3, y1, y2, y3, λ1, λ2, λ3, m}, Reals ] =!= {} &, sols, Method -> "FinestGrained", ProgressReporting -> True]; roots = Pick[sols, solQ]; MatchQ[roots, {{m -> _Root?Positive}}] ( True ) root = m /. roots[[1]] ( Root[-478425365462547737405343343 + 3773041038347596515021000956 #1 + << ... omitted for brevity ... >> 2605602600411474165760 #1^40 - 442721857769029238784 #1^41 + 147573952589676412928 #1^42 &, 11] ) Verify numerically with the previously calculated 10000 digits: res10000[[1]] - root ( 0.10^-10001) And display the value of the maximal area in a more readable way: Column[{Style[MinimalPolynomial[root, x] == 0, GrayLevel[0.15]], Row[{"near ", TildeTilde[x, N[root]]}]}, Spacings -> 0.5 {1, 1}] // TraditionalForm We can also use this method to derive the formula for maximal area unit diameter hexagon: area = 1/2 - 1/2y1 + x1y2; cons = {x1^2 + (y1 - 1)^2 - 1, (1/2 + x1)^2 + (y2 - y1)^2 - 1}; L = area - {λ1, λ2}.cons; {d1, d2, d3} = D[L, #]& /@ {x1, y1, y2}; GroebnerBasis[Join[{d1, d2, d3}, cons, {area - m}], {m}, {x1, y1, y2, λ1, λ2}] (* {11993 - 78488 m + 144464 m^2 + 1232 m^3 - 221360 m^4 + 146496 m^5 + 21056 m^6 - 30848 m^7 - 3008 m^8 + 8192 m^9 + 4096 m^10} ) Notice this is the same polynomial Graham found. Lastly, Foster, et al. (2007) tells us which vertices in the biggest little even sided n-gon are unit distance from each other. For the decagon, it looks like this and gives the following constraints and corresponding Lagrangian multiplier setup and call to `GroebnerBasis`: area = x4 - x2y1 - y2/2 + x1y2 + x2y3 - x4y3 + y4/2; cons = { (x1+x4)^2+(y1-y4)^2 == 1, x1^2+(y1-1)^2 == 1, (x2+x4)^2+(y2-y4)^2 == 1, (x2+1/2)^2+(y2-y3)^2 == 1 }; L = area - {λ1, λ2, λ3, λ4}.(Subtract @@@ cons); ds = D[L, #]& /@ {x1, x2, x4, y1, y2, y3, y4}; GroebnerBasis[ Join[ds, Subtract @@@ cons, {area - m}], {m}, {x1, x2, x4, y1, y2, y3, y4, λ1, λ2, λ3, λ4}, Method -> "GroebnerWalk" ] The output of `GroebnerBasis` is a degree-152 polynomial in `m`, one of whose roots corresponds to the maximal area. This root is approximately `0.749137`, a `1.961%` increase in area over the regular unit diameter decagon. Here is the exact form of the root: Root[ 550889780823136797957507723511972524895427242074993891937298355176558497474466417328526706272419716609053206579306974468494879127 - 41559711898973036622753555013321849453348491961662045472133332755351309435492704885947540101318827391699360592753208318810125774718#1 + 1065963350839149736288200008305070582140006819068846892409131300657379252488635326883422426434726227830299632967984821057350931506198#1^2 - 13375049592828902988764230367803264224214861704481682357510659152362505984397416660843930233056123730645026698151214182624078986714128#1^3 + 74578822372032351529299179587428053626366636762844327606991759386125665049593288636844316395337108663688987161033332032927349057508000#1^4 + 41555123203653105835292990176072324487402130809267141347506777167570357628959587525291809671105171844912298958389760307501911261288464#1^5 - 2504271069248116115349964066436843256979031869577939816643073763769332259309952361407067121375132695076852335329507885550085128753943672#1^6 + 9355226809241985949108306372748678833700684776538492415188622572682428759334816732934402676096883006169800309075381855460560170258666912#1^7 + 6785253645420863352738113620908767341749246842518706016005928952773546066744064885992777249586274767523647882393848539173726413035731512#1^8 - 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The 'largest small octagon' is the 8-sided polygon of unit diameter having largest possible area.

In 1975, Ronald Graham found a closed form expression for the area of the analogous hexagon -- a root of an irreducible degree 10 polynomial (found here).

In 2002, Audet et al. found the value of maximal octagon area to be ~0.726867 and as far as I could find, no closed form expression has been found.

Below is a picture of the maximizing hexagon and octagon (found here):

Here, the red lines indicate the vertices a unit distance from each other.

If we make some assumptions on symmetry, we can have simplified labels of the octagon's vertices and come up with constraints on them:

Our constraints will

Place bounds on location:

0 < x1 < 1/2 && 0 < x3 < 1/2 && 0 < y1 < 1/2 && 0 < y2 < 1/2 && 1/2 < y3 < 1

Add unit diameter constraints:

(x1 + x3)^2 + (y1 - y3)^2 == 1 && x1^2 + (y1 - 1)^2 == 1 && (1/2 + x3)^2 + (y2 - y3)^2 == 1

We aim to maximize area, which is given by:
```
x3 - y1/2 + x1 y2 - x3 y2 + y3/2
```

Unfortunately a (symbolic) call to Maximize did not return a result in a reasonable amount of time:

area = x3 - y1/2 + x1 y2 - x3 y2 + y3/2;
bounds = {0 < x1 < 1/2, 0 < x3 < 1/2, 0 < y1 < 1/2, 0 < y2 < 1/2, 1/2 < y3 < 1};
cons = {(x1 + x3)^2 + (y1 - y3)^2 == 1, x1^2 + (y1 - 1)^2 == 1, (1/2 + x3)^2 + (y2 - y3)^2 == 1};

Maximize[{area, Join[bounds, cons]}, {x1, x3, y1, y2, y3}]
 (* $Aborted *)

Numerical optimization works nonetheless but slows down quickly as working precision goes up:

NMaximize[{area, Join[bounds, cons]}, {x1, x3, y1, y2, y3}, WorkingPrecision -> 1000, MaxIterations -> 1000]; // AbsoluteTiming
 (* {675.194, Null} *)

To get higher precision results faster, we can do some symbolic preprocessing to reduce our parameter space:

cons2 = Reduce[And @@ Join[bounds, cons], {x1, y1, x3}, Reals, Backsubstitution -> True][[1]];
area2 = area /. ToRules[cons2 /. _Inequality -> True];

res10000 = NMaximize[{area2, cons2 /. _Equal -> True}, {y2, y3}, WorkingPrecision -> 10000]; // AbsoluteTiming
 (* {7.24073, Null} *)

Notice the ~100x speedup despite increasing the working precision itself 10x.

Since the call to Maximize didn't work, we can take matters into our own hands:

Formulate the problem through Lagrangian multipliers:

consdiff = Subtract @@@ cons;
L = area - {λ1, λ2, λ3}.consdiff;
ds = D[L, #] & /@ {x1, x3, y1, y2, y3};
cons3 = Join[ds, consdiff, {area - m}];

Use GroebnerBasis to find a polynomial relation (the magic step):

gb = GroebnerBasis[cons3, {m}, {x1, x3, y1, y2, y3, λ1, λ2, λ3}];
MatchQ[gb, {poly_} /; PolynomialExpressionQ[poly, m, NumericQ]]
 (* True *)

Select the valid root as the area maximizer, and verify it symbolically:

Length[sols = Solve[gb[[1]] == 0, m, PositiveReals]]
 (* 9 *)

solQ = ParallelMap[
  Solve[
    Join[Thread[cons3 == 0], bounds, {m == (m /. #)}],
    {x1, x3, y1, y2, y3, λ1, λ2, λ3, m},
    Reals
  ] =!= {} &, 
  sols, Method -> "FinestGrained", ProgressReporting -> True];

roots = Pick[sols, solQ];
MatchQ[roots, {{m -> _Root?Positive}}]
 (* True *)

root = m /. roots[[1]]
 (* Root[-478425365462547737405343343 + 3773041038347596515021000956 #1 + 
    << ... omitted for brevity ... >>
    2605602600411474165760 #1^40 - 442721857769029238784 #1^41 + 
    147573952589676412928 #1^42 &, 11] *)

Verify numerically with the previously calculated 10000 digits:

res10000[[1]] - root
 (* 0.*10^-10001*)

And display the value of the maximal area in a more readable way:

Column[{Style[MinimalPolynomial[root, x] == 0, GrayLevel[0.15]], 
  Row[{"near ", TildeTilde[x, N[root]]}]}, Spacings -> 0.5 {1, 1}] // TraditionalForm

enter image description here

We can also use this method to derive the formula for maximal area unit diameter hexagon:

area = 1/2 - 1/2*y1 + x1*y2;
cons = {x1^2 + (y1 - 1)^2 - 1, (1/2 + x1)^2 + (y2 - y1)^2 - 1};
L = area - {λ1, λ2}.cons;
{d1, d2, d3} = D[L, #]& /@ {x1, y1, y2};

GroebnerBasis[Join[{d1, d2, d3}, cons, {area - m}], {m}, {x1, y1, y2, λ1, λ2}]
 (* {11993 - 78488 m + 144464 m^2 + 1232 m^3 - 221360 m^4 + 
  146496 m^5 + 21056 m^6 - 30848 m^7 - 3008 m^8 + 8192 m^9 + 
  4096 m^10} *)

Notice this is the same polynomial Graham found.

Lastly, Foster, et al. (2007) tells us which vertices in the biggest little even sided n-gon are unit distance from each other. For the decagon, it looks like this

and gives the following constraints and corresponding Lagrangian multiplier setup and call to GroebnerBasis:

area = x4 - x2*y1 - y2/2 + x1*y2 + x2*y3 - x4*y3 + y4/2;
cons = {
  (x1+x4)^2+(y1-y4)^2 == 1,
  x1^2+(y1-1)^2 == 1,
  (x2+x4)^2+(y2-y4)^2 == 1,
  (x2+1/2)^2+(y2-y3)^2 == 1
};

L = area - {λ1, λ2, λ3, λ4}.(Subtract @@@ cons);
ds = D[L, #]& /@ {x1, x2, x4, y1, y2, y3, y4};

GroebnerBasis[
  Join[ds, Subtract @@@ cons, {area - m}],
  {m}, {x1, x2, x4, y1, y2, y3, y4, λ1, λ2, λ3, λ4},
  Method -> "GroebnerWalk"
]

The output of GroebnerBasis is a degree-152 polynomial in m, one of whose roots corresponds to the maximal area. This root is approximately 0.749137, a 1.961% increase in area over the regular unit diameter decagon.

Here is the exact form of the root:

Root[
    550889780823136797957507723511972524895427242074993891937298355176558497474466417328526706272419716609053206579306974468494879127
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  - 13375049592828902988764230367803264224214861704481682357510659152362505984397416660843930233056123730645026698151214182624078986714128*#1^3
  + 74578822372032351529299179587428053626366636762844327606991759386125665049593288636844316395337108663688987161033332032927349057508000*#1^4
  + 41555123203653105835292990176072324487402130809267141347506777167570357628959587525291809671105171844912298958389760307501911261288464*#1^5
  - 2504271069248116115349964066436843256979031869577939816643073763769332259309952361407067121375132695076852335329507885550085128753943672*#1^6
  + 9355226809241985949108306372748678833700684776538492415188622572682428759334816732934402676096883006169800309075381855460560170258666912*#1^7
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  - 245611452956611669128047930483404983007190274011915098010367804970152429897878927664525417397060960727160021733013846609531072652453251008*#1^11
  - 856739812895278063251757637852645639375329750017913620468072798195499499715617755305718071355956375517217645747601279517517676292055669472*#1^12
  + 10578585938800364043736225963493466308735483576190618410045168374584740399018871324203209493611233487958582072271867720454440342028837648128*#1^13
  - 37852620470100605721909443390470977345936696639358159684071265106850246421876208042059403089502550062582563437000071587973831516396570543616*#1^14
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  + 432091534389039000357992567317779225969741839936554352292492094340383657462510459899817272349232455701106080208198675117608169176288644703232*#1^16
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  - 1593441786201215142815293155666172060187112861347202616428691342689918278670240125758196885682154831143521183493979763032318099697977486796800*#1^18
  + 8171459385377387023346721286133893945521740593935633729610203045851035037666842952246214908566616456534579785911025333688124303152606231355392*#1^19
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  - 260418162780805562933152457341477004462982083673459068728280905297658481895031978813646560034816*#1^149
  + 31517467983489796598020772129638039053269935528347092537659132354676126773362676324916847968256*#1^150
  - 603671764361675829561034250937498855483700923339941277251139951100346133941422867830137683968*#1^151
  + 94078716523897531879641701444805016439018325715315523727450381989664332562299667713787691008*#1^152&,
  24
]

POSTED BY: Greg Hurst

3 Replies

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Ed Pegg

Ed Pegg, Wolfram Research

Posted 2 days ago

Beyond that, the biggest little polyhedra: https://blog.wolfram.com/2015/05/20/biggest-little-polyhedronnew-solutions-in-combinatorial-geometry/

POSTED BY: Ed Pegg

Bernd Mulansky

Posted 6 hours ago

I just want to mention that the closed form expression of $A_8^*$ as a root of an integer polynomial of degree $42$ is given by Audet, Hansen, Svrtan 2021 in https://link.springer.com/article/10.1007/s10898-020-00908-w The case $n=10$ is contained in https://tinyurl.com/c7hpb752 The approach using Lagrange multipliers is also sketched in https://arxiv.org/abs/2404.01841 , where a representation of the maximum perimeter of a small octagon is derived.

POSTED BY: Bernd Mulansky

EDITORIAL BOARD

EDITORIAL BOARD, WOLFRAM

Posted 4 days ago

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POSTED BY: EDITORIAL BOARD

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