How can I create a minimal surface with trefoil knot as inner edge and circle as outer edge?

Question

This minimal surface is the subject of Michael Foster's beautiful wood-turned sculpture "Inversion" (see http://breezyhillturning.com/styled-2/photos-2/index.html).

In addition, how can one create 2D projections of this minimal surface?

Answer 1

Let's start with a very coarse triangle mesh that is spanned into coarse approximations of both a round circle and a trefoil knot:

R0 = MeshRegion[
   {{0.7500273036793702`, 3.76147758203428`*^-17, 2.906278119369538`*^-17}, {0.8705378492638793`, 0.07751720292285104`, 0.0008837176573538955`}, {0.7713524614216626`, 0.33636054838561097`, 0.006228199166873891`}, {0.6573509103985639`, 0.3070512205130426`, 0.028588437239951532`}, {0.6610310804045564`, -0.2984155385454792`, -0.029655789809020023`}, {0.8330929515604932`,  -0.2283642507965738`, -0.002046402312729532`}, {0.6791661090527485`, 0.002270249212695186`, -0.0002805981436901225`},  {0.7277435204772187`, -0.4206035972270632`, -0.00833424354619048`}, {-0.3750136518396851`, 0.6495426985182803`, -2.3629795066625606`*^-17},  {-0.502400791593442`, 0.7151492909569623`, 0.0008837176573538783`}, {-0.6769730104436353`, 0.4998305526700104`, 0.006228199166873873`}, {-0.5945896124265945`, 0.41575697734946326`, 0.028588437239951497`}, {-0.07208010293787873`, 0.7216774775941591`, -0.029655789809020165`},  {-0.218777233274213`, 0.8356617851634327`, -0.0020464023127295563`},  {-0.3415491480174897`, 0.5870399792227651`, -0.00028059814369024353`},  {0.0003816398831454383`, 0.8405461747863238`, -0.008334243546190527`},  {-0.3750136518396851`, -0.6495426985182804`, 3.9668187899865235`*^-18}, {-0.3681370576704374`,  -0.7926664938798135`, 0.0008837176573538869`}, {-0.0943794509780274`,  -0.8361911010556216`, 0.0062281991668738795`}, {-0.06276129797196948`, -0.7228081978625059`, 0.028588437239951504`}, {-0.5889509774666775`, -0.4232619390486799`, -0.02965578980902008`}, {-0.6143157182862803`, -0.6072975343668591`, -0.002046402312729543`}, {-0.33761696103525873`, -0.5893102284354601`, -0.00028059814369018016`}, {-0.7281251603603641`, -0.4199425775592605`, -0.008334243546190499`}, {-0.0000573538651297449`, -0.0021211046948096464`, 0.010235243567421528`}, {-0.11515098290606908`, -0.10323214348312777`, 0.03651746930822393`}, {0.08003195228032729`, -0.1389576306413695`, 0.0015590551459508308`}, {0.14919313946910776`, 0.04539711796317642`, -0.041715744314098344`}, {-0.03469928049253633`, 0.15563767609535087`, 0.03148964601765504`}, {0.25000000000000006`, 0.4330127018922193`, 0.25`}, {0.25`, 0.4330127018922197`, -0.25`}, {0.2500000000000001`, -0.4330127018922194`, 0.25`}, {0.24999999999999925`, -0.433012701892219`, -0.25`}, {-0.5000000000000001`, 1.8369701987210302`*^-16, 0.25`}, {-0.49999999999999994`, 6.123233995736765`*^-17, -0.25`}, {0.7071067811865474`, -0.7071067811865477`, 0.`}, {0.8660254037844384`, -0.5000000000000004`, 0.`}, {0.9659258262890683`, -0.2588190451025207`, 0.`}, {1.`, 0.`, 0.`}, {0.9659258262890683`, 0.25881904510252074`, 0.`}, {0.8660254037844387`, 0.49999999999999994`, 0.`}, {0.7071067811865476`, 0.7071067811865475`, 0.`}, {0.75`, 0.`, 0.`}, {0.25881904510252074`, 0.9659258262890683`, 0.`}, {6.123233995736766`*^-17, 1.`, 0.`}, {-0.25881904510252085`, 0.9659258262890683`, 0.`}, {-0.4999999999999998`, 0.8660254037844387`, 0.`}, {-0.7071067811865475`, 0.7071067811865476`, 0.`}, {-0.8660254037844387`, 0.49999999999999994`, 0.`}, {-0.9659258262890682`, 0.258819045102521`, 0.`}, {-0.37499999999999944`, 0.6495190528383293`, -1.2246467991473532`*^-16}, {-0.9659258262890684`, -0.25881904510252035`, 0.`}, {-0.8660254037844386`, -0.5000000000000001`, 0.`}, {-0.7071067811865477`, -0.7071067811865475`, 0.`}, {-0.5000000000000004`, -0.8660254037844384`, 0.`}, {-0.25881904510252063`, -0.9659258262890683`, 0.`}, {-1.8369701987210297`*^-16, -1.`, 0.`}, {0.2588190451025203`, -0.9659258262890684`, 0.`}, {-0.37500000000000033`, -0.6495190528383288`, -6.123233995736766`*^-17}, {-0.12499999999999994`, 0.21650635094610968`, 3.061616997868383`*^-17}, {-0.12499999999999996`, -0.21650635094610968`, 1.5308084989341916`*^-16}, {0.25`, 0.3383883476483184`, 0.17677669529663687`}, {1.9791719275740164`*^-17, 0.32322330470336313`, 0.1767766952966369`}, {0.2799195929682706`, 0.16161165235168123`, -0.17677669529663664`},  {2.0720310340813742`*^-16, 0.6767766952966366`, -0.17677669529663712`}, {-1.243218620448825`*^-16, -0.676776695296637`, 0.17677669529663684`}, {0.2799195929682708`, -0.16161165235168182`, 0.1767766952966371`}, {-1.3854203493018116`*^-16, -0.32322330470336313`, -0.17677669529663692`}, {0.5861058108161683`, -0.33838834764831854`, -0.17677669529663653`}, {-0.586105810816168`, 0.3383883476483182`, 0.17677669529663695`}, {-0.27991959296827107`, -0.1616116523516814`, 0.17677669529663714`}, {-0.2799195929682707`, 0.1616116523516815`, -0.17677669529663687`}, {-0.586105810816168`, -0.33838834764831854`, -0.17677669529663675`}, {0.5000000000000001`, 0.8660254037844387`, 0.`}, {0.5000000000000001`, -0.8660254037844387`, 0.`}, {-1.`, 1.2246467991473532`*^-16, 0.`}},
   Triangle[{{30, 64, 65}, {30, 65, 31}, {63, 66, 75}, {30, 31, 66}, {30, 66, 63}, {32, 68, 69}, {32, 69, 33}, {67, 70, 76}, {32, 33, 70}, {32, 70, 67}, {34, 72, 73}, {34, 73, 35}, {71, 74, 77}, {34, 35, 74}, {34, 74, 71}, {4, 3, 75}, {8, 1, 5}, {7, 1, 4}, {40, 2, 39}, {3, 2, 40}, {42, 3, 41}, {76, 5, 70}, {63, 4, 75}, {70, 5, 7}, {38, 6, 37}, {6, 8, 37}, {3, 4, 1}, {7, 63, 43}, {6, 38, 2}, {39, 2, 38}, {1, 8, 6}, {75, 3, 42}, {2, 1, 6}, {2, 3, 1}, {4, 63, 7}, {8, 76, 36}, {3, 40, 41}, {70, 7, 43}, {1, 7, 5}, {76, 8, 5}, {37, 8, 36}, {12, 11, 77}, {16, 9, 13}, {15, 9, 12}, {48, 10, 47}, {11, 10, 48}, {50, 11, 49}, {75, 13, 66}, {71, 12, 77}, {66, 13, 15}, {46, 14, 45}, {14, 16, 45}, {11, 12, 9}, {15, 71, 51}, {14, 46, 10}, {47, 10, 46}, {9, 16, 14}, {77, 11, 50}, {10, 9, 14}, {10, 11, 9}, {12, 71, 15}, {16, 75, 44}, {11, 48, 49}, {66, 15, 51}, {9, 15, 13}, {75, 16, 13}, {45, 16, 44}, {20, 19, 76}, {24, 17, 21}, {23, 17, 20}, {56, 18, 55}, {19, 18, 56}, {58, 19, 57}, {77, 21, 74}, {67, 20, 76}, {74, 21, 23}, {54, 22, 53}, {22, 24, 53}, {19, 20, 17}, {23, 67, 59}, {22, 54, 18}, {55, 18, 54}, {17, 24, 22}, {76, 19, 58}, {18, 17, 22}, {18, 19, 17}, {20, 67, 23}, {24, 77, 52}, {19, 56, 57}, {74, 23, 59}, {17, 23, 21}, {77, 24, 21}, {53, 24, 52}, {68, 27, 69}, {73, 29, 60}, {25, 29, 73}, {61, 27, 26}, {26, 72, 61}, {62, 28, 27}, {29, 28, 65}, {29, 65, 64}, {72, 26, 73}, {29, 64, 60}, {62, 27, 68}, {73, 26, 25}, {26, 27, 25}, {62, 65, 28}, {27, 28, 25}, {27, 61, 69}, {28, 29, 25}}]
  ]

Building this from

γ = t \[Function] Evaluate[1/4 KnotData[{"TorusKnot", {2, 3}}, "SpaceCurve"][t]];
δ = t \[Function] {Cos[t], Sin[t], 0};

involved already quite a lot of handcraft (solving several Possion problems and glueing pieces together).

Next, we refine this mesh utilizing the function LoopSubdivide from this post:

 R = Nest[LoopSubdivide, R0, 5]

Finally, we apply areaGradientDescent from this post:

areaGradientDescent[R]

Initial area = 2.92354

Final area = 2.7772

This is a view from the top: