Surface Fotos by Sugimoto

Exhibit: table of content

Mathematics Math21a, Fall 2005
Multivariable Calculus
Oliver Knill, SciCtr 434,

The following exposition appeared in the New York Times Magazine on Sunday, December 5, 2004. We tried the same Sunday morning to reproduce the pictures using Mathematica. The source code can be found below.

Citation NYT: "It's hard to imagine that these plaster forms, so starkly beautiful, were originally used to teach advanced students trigonometry. Called stereometric models, they were manufactured in turn-of-the-century Germany to help scholars grasp complex mathematical formulas. Last year, the Japanese photographer Hiroshi Sugimoto shot each object, the tallest of which is less than a foot high, from below at close range so that they appear monumental. His series of photographs, 'Mathematical Forms', reimagine these scientific models as things of wonder. They embody Sugimoto's belief that art is possible even without artistic intention."

Hiroshi Sugimoto is a Japanese born American Photographer. His commitment to taking photographs that combine the conceptual, the sensual and the technical has made him a rising contemporary art star.

Photo of Hiroshi Sugimoto Mathematica picture Figure capture New York Times

Mathematica Source code

 (* Rebuilding the surfaces in Mathematica which appeared in the   *) 
 (* New York Times Magazine (Photos of the American Artist         *)
 (* Hiroshi Sugimoto, December 5, 2004                             *)
 (* Oliver Knill, December 5, 2004, for Math21b, Harvard Fall 2004 *)
     ViewPoint->{0,1,0},ViewVertical -> {0,0,-1}]
 f1[u_,v_]:={Sinh[v] Cos[u],Sinh[v] Sin[u],NIntegrate[Sqrt[1-Cosh[t]^2/4],{t,0,v}]}
 f2[u_,v_]:={Sinh[v] Cos[u],Sinh[v] Sin[u],u}
 f3[u_,v_]:={Cos[u]/Cosh[v],Sin[u]/Cosh[v],v-Tan[v]+3 u}
 f4[u_,v_]:={Cos[u] Cos[v]*4/5,Sin[u] Cos[v]*4/5,
 f5[u_,v_]:={Cosh[v] Cos[u],Cosh[v] Sin[u],NIntegrate[Sqrt[1-Sinh[t]^2/4],{t,0,v}]}
 f6[u_,v_]:={2 Sqrt[1+u^2] Sin[v] Cos[u-ArcTan[u]]/(1+u^2 Sin[v]^2),
             2 Sqrt[1+u^2] Sin[v] Sin[u-ArcTan[u]]/(1+u^2 Sin[v]^2),
             Log[Tan[v/2]]+2 Cos[v]/(1+u^2 Sin[v]^2)};
 eqn = 81 (x^3 + y^3 + z^3) - 189 (x^2y + x^2z + y^2x + y^2z + z^2x + z^2y) +   
       54x y z + 126 (x y + x z + y z) - 9 1(x^2+y^2+z^2) - 9 (x+y+z)-2;
 (* eqn from *)
 S7=ContourPlot3D[eqn,{x,-1,1},{y,-1,1},{z,-1,1}, Background->GrayLevel[0.0],
Please send comments to
Oliver Knill, Math21a, Multivariable Calculus, Fall 2005, Department of Mathematics, Faculty of Art and Sciences, Harvard University