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Symbolic substitutions are introduced in C. Kimberling, "Symbolic substitutions in the transfigured plane of a triangle," Aequationes Mathematicae 73 (2007) 156-171. See also Manfred Evers, Symbolic substitution has a geometric meaning, Forum Geometricorum, vol.14 (2014), pp 217--232.

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A symbolic substitution of the form SS{a → a'} can be defined in different equivalent ways. Indeed, a' can be a function of the lengths a, b, c of the sides, or a (most of the time trigonometric) function of the angles A, B, C, or the first barycentric coordinate of a triangle center X(n) in which case we shall write SS{Xn} in short.

For instance, SS{X3} is SS{a → a^2(-a^2 + b^2 + c^2)} or SS{a → a^2 SA} or SS{a → sin 2A}.

As mentioned in ETC, these symbolic substitutions map lines to lines, conics to conics, cubics to cubics, etc. In this page, we will not examine their properties and will focus exclusively on some remarkable cubics with important contributions by Peter Moses.

N is the number of centers of the SS transform of the cubic Knnn. Two groups of cubics are particularly involved, those mentioned in pages K323 and K718.

N

Knnn

SS

X(i) on the SS transform of Knnn for i in ...

14

K1083

SS{X9}

{1, 2, 8, 9, 144, 165, 210, 518, 3057, 3307, 3308, 15587, 24646, 24647} K1084

14

K907

SS{X188}

14

K1083

SS{X3}

{2, 3, 6, 20, 69, 154, 511, 2574, 2575, 3917, 5907, 6467, 13414, 13415} K1085

12

K779

SS{X366}

{2, 7, 8, 346, 350, 3912, 4876, 5853, 14942, 14943, 18025, 20533}

pK(X8, X3912)

12

K323

SS{X8}

12

K766

SS{X7}

12

K738

SS{X508}

12

K623

SS{X9}

12

K780

SS{X188}

12

K778

SS{X366}

{2, 8, 75, 144, 518, 673, 3912, 4437, 9311, 9436, 18025, 20935}

pK(X8, X3912)

12

K770

SS{X8}

12

K767

SS{X7}

12

K356

SS{X508}

12

K718

SS{X188}

12

K777

SS{X366}

{1, 2, 7, 516, 673, 2481, 3729, 3912, 6185, 10025, 10405, 14942}

pK(X673, X14942)

12

K768

SS{X8}

12

K769

SS{X7}

12

K354

SS{X508}

12

K776

SS{X188}

12

K776

SS{X366}

{2, 8, 9, 239, 673, 2319, 2481, 4373, 5853, 9312, 9436, 14942}

pK(X14942, X673)

12

K769

SS{X8}

12

K768

SS{X7}

12

K322

SS{X508}

12

K777

SS{X188}

12

K1017

SS{X4}

{2, 4, 193, 263, 393, 3424, 6525, 6620, 6776, 7735, 9292, 9752}

pK(X6620, X4)

12

K1016

SS{Sqrt(X4)}

12

K982

SS{X9}

{1, 2, 9, 192, 518, 1280, 2340, 3158, 3693, 4876, 6168, 8299}

pK(X2340, X1)

12

K783

SS{X188}

12

K506

SS{X4}

{2, 20, 64, 235, 253, 393, 1105, 6330, 6525, 6526, 14249, 16318}

11

K718

SS{X366}

{2, 7, 85, 145, 335, 518, 3912, 9436, 10029, 14942, 16593}

pK(X9436, X7)

11

K767

SS{X8}

11

K770

SS{X7}

11

K739

SS{X508}

11

K778

SS{X188}

11

K768

SS{X10}

{2, 86, 350, 1125, 6542, 6625, 6650, 9505, 11599, 17770, 20536}

pK(X6650, X11599)

11

K623

SS{X239}

11

K769

SS{X86}

11

K251

SS{X8}

{2, 8, 75, 3161, 3693, 3717, 3912, 4518, 6376, 6559, 17755}

pK(3717, X2)

11

K996

SS{X7}

11

K984

SS{X10}

{10, 37, 42, 321, 740, 3930, 4179, 6542, 13576, 17759, 20694}

pK(X756, X740)

11

K984

SS{X37}

11

K769

SS{X10}

{2, 10, 75, 740, 1213, 6650, 9278, 11599, 17731, 20016, 24731}

pK(X11599, X6650)

11

K768

SS{X86}

11

K766

SS{X10}

{2, 10, 86, 239, 1509, 6650, 13174, 13610, 17731, 17770, 18827}

pK(X6542, X10)

11

K323

SS{X86}

11

K1009

SS{X366}

{2, 43, 55, 57, 165, 200, 1376, 2319, 3158, 8056, 19605}

pK(X55, X1376)

11

K1009

SS{X188}

11

K357

SS{X366}

{1, 2, 9, 291, 518, 2481, 3912, 6184, 9436, 14943, 17755}

pK(X518, X2)

11

K357

SS{X188}

11

K294

SS{X37}

10, 37, 42, 213, 1757, 2238, 2664, 18785, 18793, 20683, 21830}

11

K982

SS{X3}

{2, 3, 6, 194, 511, 2967, 2987, 3167, 3289, 15143, 17974}

11

K453

SS{X3}

{2, 3, 6, 30, 265, 1511, 3163, 3284, 11064, 14910, 14919}

11

K1065

SS{X188}

{1, 8, 34, 65, 85, 279, 1212, 2082, 5998, 14584, 21132}

11

K803

SS{X188}

{519, 1001, 1320, 3307, 3308, 5239, 5240, 7026, 7043, 10707, 14942}

11

K505

SS{X188}

{2, 9, 519, 908, 1000, 1320, 3872, 5239, 5240, 7026, 7043}

11

K364

SS{X188}

{4, 10, 21, 40, 78, 145, 188, 280, 3152, 3680, 13583}

11

K506

SS{X9}

{2, 7, 8, 55, 220, 480, 1223, 2340, 2346, 3059, 14942}

10

K1010

SS{X366}

{7, 9, 75, 144, 192, 346, 3161, 3729, 4373, 7155}

pK(X8, X3729)

10

K132

SS{X8}

10

K744

SS{X7}

10

K743

SS{X508}

10

K1011

SS{X9}

10

K675

SS{X188}

10

K251

SS{X4}

{2, 4, 132, 232, 264, 297, 1249, 6330, 6530, 6531}

pK(X6530, X2)

10

K252

SS{Sqrt(X4)}

10

K996

SS{X69}

10

K136

SS{X4}

{2, 4, 6, 98, 230, 419, 3224, 6530, 6531, 16081}

pK(X6531, X6531)

10

K787

SS{Sqrt(X4)}

10

K868

SS{X69}

10

K135

SS{X4}

{4, 25, 98, 297, 393, 459, 6531, 9308, 16081, 16318}

pK(?, X98)

10

K532

SS{Sqrt(X4)}

10

K994

SS{X69}

10

K534

SS{X366}

{1, 8, 188, 519, 1320, 5239, 5240, 5541, 7026, 7043}

pK(X9, X519)

10

K1080

SS{X9}

10

K001

SS{X188}

10

K996

SS{X4}

{2, 69, 76, 325, 5976, 6337, 6374, 6393, 6394, 8781}

pK(X6393, X2)

10

K251

SS{X69}

10

K343

SS{X4}

{4, 196, 459, 1249, 3176, 3183, 7003, 7149, 8894, 14361}

pK(X393, X14361)

10

K174

SS{Sqrt(X4)}

10

K1017

SS{X7}

{2, 7, 145, 279, 390, 1002, 3598, 5222, 9309, 9533}

pK(X3598, X7)

10

K1016

SS{X508}

10

K253

SS{X8}

{1, 2, 9, 346, 3452, 3680, 3752, 6552, 6736, 12640}

pK(?, X2)

10

K924

SS{X188}

10

K323

SS{X10}

{2, 10, 86, 335, 594, 740, 6542, 10026, 11599, 17762}

pK(X10, X6542)

10

K766

SS{X86}

10

K1013

SS{X18297}

{2, 75, 76, 330, 870, 871, 4441, 6063, 10009, 20917}

pK(X871, ?)

10

K1018

SS{X75}

10

K1014

SS{X366}

{1, 2, 75, 85, 87, 870, 4384, 8033, 14621, 24349}

pK(X870, X870)

10

K1014

SS{X18297}

10

K699

SS{X366}

{335, 673, 3512, 4366, 6650, 6651, 17738, 17755, 18037, 20533}

pK(239, X17738)

10

K132

SS{X239}

10

K961

SS{X239}

{350, 385, 1447, 1914, 2238, 3509, 3510, 3684, 8844, 18786}

10

K673

SS{X239}

{1, 238, 239, 350, 1914, 3253, 4366, 6654, 8299, 17475}

10

K770

SS{X894}

{2, 6, 291, 894, 1220, 1281, 3509, 7166, 8424, 17789}

10

K323

SS{X894}

{2, 257, 335, 385, 894, 3497, 3509, 6645, 17738, 17739}

10

K982

SS{X37}

{2, 10, 37, 210, 740, 1654, 2238, 9278, 13576, 16609}

10

K1079

SS{X3}

{3, 6, 219, 222, 394, 1073, 1433, 1498, 7078, 15905}

10

K769

SS{X3}

{2, 3, 69, 216, 511, 1972, 1987, 9291, 14941, 16089}

10

K510

SS{X3}

{74, 113, 265, 399, 1511, 2931, 5504, 12383, 16163, 20123}

10

K506

SS{X3}

{2, 4, 54, 69, 184, 287, 577, 1092, 3289, 5562}

10

K382

SS{X3}

{2, 3, 68, 69, 577, 1147, 3289, 3917, 9967, 23195}

10

K332

SS{X3}

{6, 69, 371, 372, 5374, 6515, 9937, 11090, 11091, 15316}

10

K323

SS{X3}

{2, 3, 264, 287, 401, 511, 577, 1988, 5374, 14941}

10

K254

SS{X3}

{2, 3, 68, 97, 264, 317, 324, 5562, 5889, 8795}

10

K168

SS{X3}

{2, 317, 371, 372, 577, 1147, 5408, 5409, 10960, 10962}

10

K058

SS{X3}

{3, 5, 30, 265, 1807, 7100, 10217, 10218, 15392, 20123}

10

K482

SS{X4}

{4, 107, 112, 648, 1294, 2479, 2480, 3163, 6531, 16080}

10

K885

SS{X188}

{2, 10, 80, 149, 519, 5239, 5240, 5692, 7026, 7043}

10

K884

SS{X188}

{2, 8, 11, 392, 519, 1145, 5239, 5240, 7026, 7043}

10

K696

SS{X8}

{1, 145, 346, 2136, 3161, 3680, 4373, 4936, 6736, 23617}

10

K982

SS{X10}

{2, 10, 37, 740, 1655, 3948, 3971, 4037, 4039, 4368}

10

K980

SS{X10}

{10, 37, 226, 335, 756, 3930, 3948, 3963, 6541, 20715}

10

K506

SS{X10}

{1, 2, 75, 335, 594, 756, 1089, 1224, 4037, 4647}

10

K1013

SS{X75}

{2, 76, 871, 1502, 2998, 3114, 7034, 10010, 18022, 20023}

10

K775

SS{X75}

{75, 76, 321, 334, 1916, 3263, 6063, 17789, 18891, 18895}

10

K423

SS{X75}

{2, 75, 76, 141, 183, 308, 327, 10010, 20917, 24273}

10

K252

SS{X75}

{2, 76, 308, 1920, 1921, 3978, 6374, 14603, 18277, 18896}

10

K778

SS{X18297}

{1, 2, 192, 726, 1575, 3551, 4876, 16557, 20671, 21219}

10

K484

SS{X366}

{1, 80, 88, 1015, 2802, 3227, 4370, 4440, 4876, 9278}

10

K482

SS{X366}

{1, 80, 88, 100, 190, 292, 644, 1018, 1120, 4370}

10

K273

SS{X366}

{1, 2, 7, 88, 519, 679, 903, 1320, 3218, 22464}

10

K239

SS{X366}

{2, 514, 519, 900, 903, 1086, 1647, 6548, 6549, 14078}