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2013年1月24日 (木)

本当のところはパラメータいくつで象の絵が描ける? -フォンノイマンとWolframAlpha

物理学者のダイソンがある理論を作って意気揚々とフェルミに会いに行ったとき、その理論に含まれるパラメータが多いことにフェルミは全然興味を持たずフォンノイマンの以下の言葉を引用したそうだ。

With four parameters I can fit an elephant, and with five I can make him wiggle his trunk.

(パラメータが4つあったら象だって描けるし、5つありゃ象の鼻を動かせるわ!)

http://en.wikiquote.org/wiki/John_von_Neumann

そのお言葉を本当にやった人がいて、、、

Drawing an elephant with four complex parameters” by Jurgen Mayer, Khaled Khairy, and Jonathon Howard, Am. J. Phys. 78, 648 (2010), DOI:10.1119/1.3254017.

GeoGebraで描いてみたりしたのだが、、、

Elephant

http://sci.tea-nifty.com/blog/2012/08/geogebra40745-0.html

hirax.net経由で、WolframAlphaは有名人の顔をパラメータ表示でグラフにするというとんでもない機能があると知った。

http://www.hirax.net/diaryweb/2013/01/24.html#10331

てことは、、、絶対に象の絵はあるはずと思って調べたら、、、

”elephant curve”で検索。

http://www.wolframalpha.com/input/?i=elephant+curve

Msp190161a50bf89h3g499c300003ffc516

象!しかしこのリンクにさらに驚くべきものが!

Benjamin the elephant-like curveを検索すると!

http://www.wolframalpha.com/input/?i=Benjamin+the+elephant-like+curve

Msp3941a50c6gbeaf8dedg000052f200170


でもこの絵をかくのに使う式というのは以下になるが、、、パラメータ4つや5つの世界じゃないですね。これはさすがにフォンノイマン、フェルミの精神から全く離れちゃってる、、、

x(t) = ((-1/30 sin(39/29-14 t)-13/16 sin(20/13-10 t)-73/26 sin(11/7-9 t)-215/26 sin(58/37-8 t)-105/16 sin(25/16-6 t)+384/5 sin(t+11/7)+621/31 sin(2 t+80/17)+551/25 sin(3 t+58/37)+425/31 sin(4 t+80/17)+703/68 sin(5 t+39/25)+380/39 sin(7 t+146/31)+77/13 sin(11 t+36/23)+11/20 sin(12 t+52/33)+6/31 sin(13 t+44/23)+26/27 sin(15 t+29/18)+27/16 sin(16 t+25/16)+97/28 sin(17 t+11/7)+21/16 sin(18 t+19/12)+47/35 sin(19 t+27/17)+1/29 sin(20 t+85/23)+2/5 sin(21 t+27/17)+1/21 sin(22 t+60/13)+7087/24) theta(95 pi-t) theta(t-91 pi)+(-41/51 sin(39/25-2 t)+711/23 sin(t+11/7)+93/37 sin(3 t+11/7)-1647/20) theta(91 pi-t) theta(t-87 pi)+(-71/17 sin(47/30-3 t)-73/36 sin(45/29-2 t)-1811/45 sin(58/37-t)-1994/19) theta(87 pi-t) theta(t-83 pi)+(-28/3 sin(58/37-11 t)-187/31 sin(14/9-7 t)-223/15 sin(36/23-5 t)-686/27 sin(36/23-3 t)+531/26 sin(t+11/7)+2234/41 sin(2 t+63/40)+455/22 sin(4 t+30/19)+5/2 sin(6 t+80/17)+101/28 sin(8 t+30/19)+114/17 sin(9 t+25/16)+102/29 sin(10 t+127/27)+98/13 sin(12 t+11/7)+54/25 sin(13 t+17/11)+35/24 sin(14 t+93/20)+60/37 sin(15 t+23/15)+1/9 sin(16 t+17/9)+31/32 sin(17 t+117/25)+69/16 sin(18 t+36/23)+8/33 sin(19 t+39/28)-12315/31) theta(83 pi-t) theta(t-79 pi)+(-55/23 sin(36/23-2 t)+101/7 sin(t+11/7)+28/23 sin(3 t+63/40)-10996/25) theta(79 pi-t) theta(t-75 pi)+(-47/13 sin(36/23-3 t)-1169/30 sin(11/7-t)+45/7 sin(2 t+11/7)+7/25 sin(4 t+19/12)-6701/19) theta(75 pi-t) theta(t-71 pi)+(379/27 sin(t+11/7)-596/9) theta(71 pi-t) theta(t-67 pi)+(351/23 sin(t+11/7)+69/43 sin(2 t+11/7)+93/47 sin(3 t+11/7)-2575/11) theta(67 pi-t) theta(t-63 pi)+(-129/46 sin(26/17-8 t)-237/26 sin(39/25-5 t)-284/25 sin(58/37-4 t)-92/31 sin(31/20-2 t)+884/21 sin(t+11/7)+38/37 sin(3 t+21/13)+153/14 sin(6 t+49/31)+233/18 sin(7 t+30/19)+95/16 sin(9 t+35/22)+37/23 sin(10 t+14/9)+33/20 sin(11 t+35/22)+7/17 sin(12 t+32/19)+1/33 sin(13 t+3/10)-766/15) theta(63 pi-t) theta(t-59 pi)+(-5/14 sin(11/8-14 t)-20/27 sin(26/17-13 t)-213/11 sin(17/11-5 t)-233/20 sin(17/11-4 t)+681/16 sin(t+41/26)+142/23 sin(2 t+108/23)+159/28 sin(3 t+46/29)+87/19 sin(6 t+27/17)+31/15 sin(7 t+62/37)+33/20 sin(8 t+14/9)+31/20 sin(9 t+47/10)+124/23 sin(10 t+45/28)+28/25 sin(11 t+75/16)+25/14 sin(12 t+43/27)-3889/20) theta(59 pi-t) theta(t-55 pi)+(-503/32 sin(58/37-2 t)+869/9 sin(t+11/7)+238/37 sin(3 t+11/7)+5653/25) theta(55 pi-t) theta(t-51 pi)+(-6/19 sin(47/30-7 t)-2/7 sin(47/30-6 t)-12/19 sin(53/34-5 t)-12/7 sin(36/23-4 t)+79/5 sin(t+11/7)+61/21 sin(2 t+30/19)+91/16 sin(3 t+30/19)+31/22 sin(8 t+27/17)+7/22 sin(9 t+53/33)+17/42 sin(10 t+35/22)-253/2) theta(51 pi-t) theta(t-47 pi)+(-34/19 sin(11/7-4 t)-38/23 sin(14/9-2 t)+357/19 sin(t+11/7)+77/18 sin(3 t+11/7)+86/49 sin(5 t+113/24)+29/44 sin(6 t+30/19)+3/10 sin(7 t+37/24)+23/32 sin(8 t+14/9)+19/21 sin(9 t+25/16)+2/23 sin(10 t+53/35)-5402/23) theta(47 pi-t) theta(t-43 pi)+(-1/18 sin(23/21-29 t)-1/18 sin(25/24-25 t)-119/60 sin(39/25-11 t)+911/10 sin(t+11/7)+5290/33 sin(2 t+47/30)+368/13 sin(3 t+80/17)+516/47 sin(4 t+36/23)+64/9 sin(5 t+26/17)+139/35 sin(6 t+27/17)+709/71 sin(7 t+47/10)+14/3 sin(8 t+14/9)+40/19 sin(9 t+47/10)+19/24 sin(10 t+65/14)+638/213 sin(12 t+39/25)+64/29 sin(13 t+47/10)+20/27 sin(14 t+36/23)+8/15 sin(15 t+25/17)+37/23 sin(16 t+17/11)+5/8 sin(17 t+47/10)+24/19 sin(18 t+51/32)+12/19 sin(19 t+10/7)+10/7 sin(20 t+19/12)+1/139 sin(21 t+19/8)+37/28 sin(22 t+36/23)+36/73 sin(23 t+18/13)+182/183 sin(24 t+30/19)+11/9 sin(26 t+14/9)+18/23 sin(27 t+172/115)+9/26 sin(28 t+31/20)+5/4 sin(30 t+151/101)-6302/19) theta(43 pi-t) theta(t-39 pi)+(-11/26 sin(20/13-13 t)-6/17 sin(7/5-9 t)-79/39 sin(35/23-7 t)-47/27 sin(166/111-5 t)+5780/41 sin(t+47/30)+217/33 sin(2 t+58/37)+329/29 sin(3 t+80/17)+531/25 sin(4 t+25/16)+326/29 sin(6 t+25/16)+7/16 sin(8 t+32/7)+26/21 sin(10 t+14/9)+93/47 sin(11 t+47/31)+55/32 sin(12 t+31/20)+6/35 sin(14 t+114/25)+47/63 sin(15 t+61/13)+2/7 sin(16 t+19/13)+11/25 sin(17 t+29/19)-8061/22) theta(39 pi-t) theta(t-35 pi)+(-5/24 sin(28/19-33 t)-193/19 sin(53/34-13 t)+12599/50 sin(t+127/27)+4661/42 sin(2 t+47/10)+1436/31 sin(3 t+108/23)+521/20 sin(4 t+369/79)+970/27 sin(5 t+75/16)+863/16 sin(6 t+20/13)+849/35 sin(7 t+107/23)+113/17 sin(8 t+124/27)+811/20 sin(9 t+3/2)+805/37 sin(10 t+60/13)+234/19 sin(11 t+83/18)+124/27 sin(12 t+30/19)+25/23 sin(14 t+41/11)+638/21 sin(15 t+23/5)+6/5 sin(16 t+37/15)+115/41 sin(17 t+43/36)+75/16 sin(18 t+58/37)+151/16 sin(19 t+32/7)+20/21 sin(20 t+63/25)+157/26 sin(21 t+373/80)+24/11 sin(22 t+27/16)+107/22 sin(23 t+14/3)+56/19 sin(24 t+13/8)+133/18 sin(25 t+147/32)+62/13 sin(26 t+80/53)+83/26 sin(27 t+83/18)+31/23 sin(28 t+41/26)+184/39 sin(29 t+9/7)+52/23 sin(30 t+86/19)+89/22 sin(31 t+89/20)+88/27 sin(32 t+58/13)+7/3 sin(34 t+13/10)+83/22 sin(35 t+89/67)+5/21 sin(36 t+103/22)+1394/19) theta(35 pi-t) theta(t-31 pi)+(-117/20 sin(13/9-17 t)-57/20 sin(44/29-15 t)-509/28 sin(142/95-9 t)-71/13 sin(77/58-8 t)-140/3 sin(71/47-7 t)+5129/30 sin(t+30/19)+647/54 sin(2 t+23/14)+1862/37 sin(3 t+8/5)+397/33 sin(4 t+31/19)+108/25 sin(5 t+17/10)+107/8 sin(6 t+30/19)+1256/251 sin(10 t+27/17)+161/31 sin(11 t+27/16)+9/20 sin(12 t+45/13)+273/19 sin(13 t+17/10)+101/22 sin(14 t+16/9)+65/23 sin(16 t+45/28)+16/11 sin(18 t+44/27)+1/17 sin(19 t+89/19)+19/12 sin(20 t+52/31)+75/31 sin(21 t+19/10)+3/20 sin(22 t+64/19)+28/83 sin(23 t+19/10)+2/21 sin(24 t+45/14)+9/28 sin(25 t+73/17)+37/17 sin(26 t+68/37)+24/29 sin(27 t+79/45)+6/25 sin(28 t+37/15)-1722/17) theta(31 pi-t) theta(t-27 pi)+(-25/11 sin(14/9-2 t)+421/23 sin(t+9/5)-3714/25) theta(27 pi-t) theta(t-23 pi)+(-5/21 sin(23/15-5 t)-31/19 sin(14/29-2 t)+79/29 sin(t+4)+83/22 sin(3 t+53/52)+11/13 sin(4 t+8/13)-2049/16) theta(23 pi-t) theta(t-19 pi)+(-102/47 sin(46/37-4 t)-94/23 sin(9/8-3 t)-26/27 sin(19/30-2 t)+79/27 sin(t+13/35)+47/28 sin(5 t+7/4)-4053/17) theta(19 pi-t) theta(t-15 pi)+(470/21 sin(t+7/10)-18347/80) theta(15 pi-t) theta(t-11 pi)+(738/29 sin(t+5/19)-3865/33) theta(11 pi-t) theta(t-7 pi)+(1175/14 sin(t+4/9)+9856/27) theta(7 pi-t) theta(t-3 pi)+(2229/25 sin(t+13/23)-27247/63) theta(3 pi-t) theta(t+pi)) theta(sqrt(sgn(sin(t/2))))
y(t) = ((-16/19 sin(39/25-17 t)-54/53 sin(31/20-15 t)-50/21 sin(14/9-10 t)-28/5 sin(36/23-9 t)+1641/14 sin(t+58/37)+9/13 sin(2 t+46/29)+807/29 sin(3 t+80/17)+421/26 sin(4 t+47/30)+25/14 sin(5 t+36/23)+47/25 sin(6 t+45/29)+11/32 sin(7 t+25/17)+175/24 sin(8 t+113/24)+41/16 sin(11 t+146/31)+49/26 sin(12 t+14/9)+1/11 sin(13 t+10/7)+43/32 sin(14 t+108/23)+23/12 sin(16 t+113/24)+4/11 sin(18 t+134/89)+5/21 sin(19 t+113/24)+7/36 sin(20 t+51/32)+19/48 sin(21 t+41/26)+17/33 sin(22 t+80/17)-4105/16) theta(95 pi-t) theta(t-91 pi)+(-169/8 sin(11/7-t)+40/41 sin(2 t+80/17)+88/27 sin(3 t+146/31)+13344/25) theta(91 pi-t) theta(t-87 pi)+(741/29 sin(t+11/7)+235/78 sin(2 t+113/24)+73/23 sin(3 t+11/7)+6701/13) theta(87 pi-t) theta(t-83 pi)+(-5/22 sin(17/12-16 t)-101/46 sin(47/30-5 t)-191/18 sin(25/16-4 t)-129/4 sin(36/23-3 t)-753/17 sin(58/37-2 t)-2371/26 sin(11/7-t)+4/13 sin(6 t+93/20)+23/18 sin(7 t+61/13)+127/17 sin(8 t+113/24)+2/19 sin(9 t+49/24)+53/12 sin(10 t+52/33)+97/39 sin(11 t+47/10)+80/23 sin(12 t+80/17)+29/18 sin(13 t+52/33)+sin(14 t+37/23)+1/18 sin(15 t+86/19)+49/73 sin(17 t+145/31)+13/29 sin(18 t+43/27)+57/35 sin(19 t+11/7)-1045/9) theta(83 pi-t) theta(t-79 pi)+(469/32 sin(t+58/37)+66/31 sin(2 t+39/25)+63/38 sin(3 t+36/23)+4839/17) theta(79 pi-t) theta(t-75 pi)+(-59/19 sin(36/23-4 t)-395/28 sin(11/7-2 t)+131/5 sin(t+33/7)+sin(3 t+36/23)+10448/25) theta(75 pi-t) theta(t-71 pi)+(9459/22-294/13 sin(58/37-t)) theta(71 pi-t) theta(t-67 pi)+(-43/16 sin(11/7-2 t)+181/15 sin(t+11/7)+7/18 sin(3 t+11/7)+21478/47) theta(67 pi-t) theta(t-63 pi)+(-21/11 sin(26/17-12 t)-13/32 sin(37/25-9 t)-147/13 sin(25/16-6 t)-192/17 sin(36/23-5 t)-191/32 sin(14/9-3 t)-159/11 sin(47/30-t)+557/76 sin(2 t+41/26)+136/23 sin(4 t+19/12)+19/20 sin(7 t+29/18)+14/13 sin(8 t+43/27)+11/16 sin(10 t+14/9)+141/20 sin(11 t+19/12)+34/25 sin(13 t+21/13)-10779/31) theta(63 pi-t) theta(t-59 pi)+(-7/16 sin(41/28-14 t)-17/11 sin(20/13-13 t)-13/10 sin(25/17-10 t)-124/19 sin(23/15-9 t)-335/24 sin(17/11-6 t)-85/19 sin(29/19-5 t)+251/19 sin(t+30/19)+139/36 sin(2 t+35/22)+62/9 sin(3 t+27/17)+3/7 sin(4 t+19/11)+23/15 sin(7 t+27/17)+27/23 sin(8 t+127/27)+35/18 sin(11 t+80/17)+35/17 sin(12 t+34/21)-6517/20) theta(59 pi-t) theta(t-55 pi)+(-853/20 sin(58/37-2 t)-4133/72 sin(11/7-t)+58/15 sin(3 t+19/12)+6577/24) theta(55 pi-t) theta(t-51 pi)+(-17/21 sin(25/16-9 t)-63/20 sin(11/7-5 t)-193/64 sin(11/7-2 t)+9/31 sin(t+103/22)+149/21 sin(3 t+30/19)+210/19 sin(4 t+30/19)+19/11 sin(6 t+46/29)+1/26 sin(7 t+13/3)+5/9 sin(8 t+39/25)+17/42 sin(10 t+41/26)+9751/26) theta(51 pi-t) theta(t-47 pi)+(-11/25 sin(14/9-9 t)-7/19 sin(97/65-5 t)+3/17 sin(t+14/9)+72/13 sin(2 t+11/7)+23/19 sin(3 t+29/19)+164/13 sin(4 t+58/37)+39/25 sin(6 t+113/24)+2/7 sin(7 t+45/29)+23/20 sin(8 t+39/25)+6/11 sin(10 t+47/30)+9089/23) theta(47 pi-t) theta(t-43 pi)+(-31/37 sin(20/13-13 t)+1064/13 sin(t+146/31)+1044/23 sin(2 t+113/24)+157/4 sin(3 t+36/23)+223/28 sin(4 t+58/37)+29/8 sin(5 t+27/17)+67/7 sin(6 t+45/29)+83/30 sin(7 t+14/9)+26/53 sin(8 t+65/14)+67/22 sin(9 t+43/28)+19/16 sin(10 t+36/23)+68/19 sin(11 t+89/19)+61/35 sin(12 t+23/15)+208/49 sin(14 t+131/28)+41/30 sin(15 t+61/13)+5/17 sin(16 t+85/57)+93/26 sin(17 t+89/19)+107/40 sin(18 t+159/34)+4/19 sin(19 t+32/21)+13/14 sin(20 t+75/16)+25/12 sin(21 t+345/74)+4/15 sin(22 t+108/23)+1/71 sin(23 t+58/35)+26/19 sin(24 t+289/62)+13/18 sin(25 t+149/32)+4/23 sin(26 t+13/9)+5/16 sin(27 t+127/27)+17/21 sin(28 t+257/55)+36/43 sin(29 t+93/20)+3/16 sin(30 t+117/25)+5698/19) theta(43 pi-t) theta(t-39 pi)+(-11/31 sin(71/47-13 t)+2191/34 sin(t+113/24)+78/7 sin(2 t+11/7)+967/35 sin(3 t+80/17)+131/20 sin(4 t+103/22)+193/13 sin(5 t+47/10)+12/5 sin(6 t+43/27)+71/18 sin(7 t+17/11)+91/92 sin(8 t+34/21)+15/17 sin(9 t+80/17)+49/17 sin(10 t+163/35)+35/9 sin(11 t+75/16)+7/12 sin(12 t+31/19)+75/47 sin(14 t+48/31)+13/27 sin(15 t+61/13)+54/55 sin(16 t+65/14)+32/19 sin(17 t+107/23)+6620/33) theta(39 pi-t) theta(t-35 pi)+(8498/29 sin(t+25/16)+5393/42 sin(2 t+108/23)+788/23 sin(3 t+14/3)+1776/31 sin(4 t+103/22)+49/22 sin(5 t+29/17)+311/32 sin(6 t+47/10)+909/29 sin(7 t+29/19)+238/19 sin(8 t+134/29)+361/43 sin(9 t+73/49)+1123/28 sin(10 t+28/19)+19/2 sin(11 t+275/59)+69/13 sin(12 t+46/31)+196/23 sin(13 t+53/38)+173/15 sin(14 t+27/19)+14/23 sin(15 t+19/27)+25/11 sin(16 t+14/3)+241/40 sin(17 t+119/26)+125/14 sin(18 t+91/20)+192/55 sin(19 t+105/23)+68/15 sin(20 t+67/15)+122/23 sin(21 t+119/27)+155/18 sin(22 t+76/17)+202/81 sin(23 t+81/19)+113/41 sin(24 t+71/16)+65/16 sin(25 t+61/14)+59/11 sin(26 t+89/20)+6/7 sin(27 t+78/19)+62/27 sin(28 t+32/7)+16/9 sin(29 t+139/31)+32/27 sin(30 t+32/7)+10/49 sin(31 t+8/17)+21/13 sin(32 t+49/41)+51/32 sin(33 t+20/19)+50/29 sin(34 t+13/11)+49/22 sin(35 t+142/31)+43/21 sin(36 t+123/28)+6044/19) theta(35 pi-t) theta(t-31 pi)+(-8/9 sin(21/22-28 t)-77/51 sin(19/13-26 t)-19/5 sin(19/15-25 t)-53/24 sin(49/48-24 t)-20/11 sin(7/5-23 t)-29/16 sin(27/23-21 t)-19/13 sin(48/41-20 t)-58/17 sin(53/37-19 t)-16/3 sin(35/24-17 t)-37/30 sin(11/16-16 t)-265/28 sin(45/32-13 t)-41/7 sin(13/9-12 t)-181/23 sin(53/37-9 t)-729/17 sin(50/33-8 t)-5/8 sin(23/16-2 t)-380/13 sin(58/37-t)+37/18 sin(3 t+251/58)+2322/25 sin(4 t+8/5)+378/29 sin(5 t+105/23)+2804/21 sin(6 t+34/21)+19/15 sin(7 t+16/7)+218/17 sin(10 t+13/8)+8/13 sin(11 t+31/11)+27/14 sin(14 t+68/39)+83/33 sin(15 t+149/32)+29/13 sin(18 t+45/28)+19/22 sin(22 t+29/19)+5/9 sin(27 t+49/38)-1228/7) theta(31 pi-t) theta(t-27 pi)+(416/33 sin(t+119/40)+275/59 sin(2 t+191/82)+5033/29) theta(27 pi-t) theta(t-23 pi)+(69/16 sin(t+81/28)+13/5 sin(2 t+2/23)+119/19 sin(3 t+55/21)+32/25 sin(4 t+13/22)+5/11 sin(5 t+73/26)+5435/16) theta(23 pi-t) theta(t-19 pi)+(113/27 sin(t+26/23)+88/41 sin(2 t+118/79)+62/15 sin(3 t+19/39)+23/10 sin(4 t+11/26)+73/30 sin(5 t+85/28)+8305/23) theta(19 pi-t) theta(t-15 pi)+(10874/31-4059/145 sin(23/27-t)) theta(15 pi-t) theta(t-11 pi)+(11201/34-4852/147 sin(26/19-t)) theta(11 pi-t) theta(t-7 pi)+(-3895/34 sin(15/19-t)-10588/39) theta(7 pi-t) theta(t-3 pi)+(-3568/31 sin(23/18-t)-2969/15) theta(3 pi-t) theta(t+pi)) theta(sqrt(sgn(sin(t/2))))

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