Mathematics is a naturally-evolved human language whose purpose is to describe abstraction. An abstract model of the world from the point of view of an individual is fundamentally mathematical in nature, even though that individual may not think of it as such.
Modernism established the philosophical importance of abstract, pure form and implied that pure form is mathematically describable. The advantage we have today, over the modernists of fifty to seventy-five years ago, is that our contemporary set of tools for abstract analysis is much more powerful and sophisticated than ever before.
This work posits the proposition that ultimate exaltation of the idea of the human form is a mathematical model thereof.
This image is a computer rendering of a full-body 3-D digital figure scan done by Cyberware.
A complete, 3-D digitization of the human body, such as that depicted here, can be interactively viewed, moved and examined in a visual computer. It can be viewed in stereo, given appropriate viewing hardware.
Such a 3-D data set is simultaneously a sculpture in numerical form and an abstract representation of the scanned subject ready for computer analysis and manipulation.
3-D digital scanning is a radically new means of apprehending the subject. It is a step toward the complete abstract modeling of the physique of the person who was scanned.
The work presented here proposes to complete this abstract model.
The first step is to notice that the full dataset is composed of a copius quantity of numbers with virtually no high-level information about what these numbers mean:
figure-1.obj: v 0.778582 -24.747299 1.364960 v 0.629679 -24.745199 1.456490 v 0.770956 -24.750099 1.247000 ...skipping 324,433 lines... v 1.139560 17.218800 -3.935180 v 1.144790 17.215500 -3.936950 f 1 2 3 f 4 5 6 f 7 8 9 ...skipping 108,140 lines... f 324430 324431 324432 f 324433 324434 324435 f 324436 324437 324438
The digital model approximates the smooth figure of the subject by a mesh of 108,146 polygons drawn between 324,438 three-dimensional coordinates in space.
I have deconstructed the 3-D scan dataset in several ways using some of the command-line pieces of GeomView by the Geometry Center at the University of Minnesota.
obj2off.pl figure-1.obj | polymerge -EF | clip -v 1,0,0 -s 12 | anytooff | off2obj.pl > figure-1-x-slivered_x12.obj
The preceding produces a version of the original digital model, sliced into vertical sections.
I have developed software which will extract the boundary curves from a surface sliced in this way.
Similarly, curves which trace the intersection of the figure with parallel planes can be generated in any position and orientation within the space of the figure.
obj2off.pl figure-1.obj | polymerge -EF | clip -v 0,1,0 -s 12 | anytooff | off2obj.pl > figure-1-y-slivered_x12.obj
To each of these curves, a parametric equation can be fitted.
The curve, shown above in Open Inventor, is converted to data readable by Mathematica:
<< figure-1_bernoulli_geom.m ;
Show[Graphics3D[Line[bernoulliPoints],
ViewPoint -> {1.30000000000000004`, 2.39999999999999991`, 2.`}]]
![[Graphics:Bernoulli_Approxgr1.gif]](Bernoulli_Approxgr1.gif)
- Graphics3D -
BernPtsXpose = Transpose[bernoulliPoints] ;
Xpoints = BernPtsXpose[[1]] ;
Dimensions[Xpoints]
{339}
Ypoints = BernPtsXpose[[2]] ;
Dimensions[Ypoints]
{339}
Zpoints = BernPtsXpose[[3]] ;
<< NumericalMath`PolynomialFit`
fX = PolynomialFit[Xpoints, 12]
FittingPolynomial[<>, 12 ]
Expand[fX[x]]
-0.147842169168174386` + 0.226258997669756745` x -
2 3
0.0224064337808843561` x + 0.0011146752909809483` x -
4 5
0.0000278260920029094638` x + 3.99820144022085877`*^-7 x -
6 7
3.61408967191505148`*^-9 x + 2.15347653500410141`*^-11 x -
8 9
8.61073546195163963`*^-14 x + 2.29309029794464391`*^-16 x -
10 11
3.90933599089718484`*^-19 x + 3.86619209676075605`*^-22 x
12
- 1.6887665895061259`*^-25 x
fY = PolynomialFit[Ypoints, 12]
FittingPolynomial[<>, 12 ]
Expand[fY[y]]
4.79464446349426864` - 0.161599177535849625` y +
2 3
0.0268921074388652403` y - 0.00166211728667394931` y +
4 5
0.0000506444567517341237` y - 8.76865895448001886`*^-7 y +
6 7
9.40457688304197247`*^-9 y - 6.55007756008115471`*^-11 y +
8 9
3.01978632406849145`*^-13 y - 9.15857653791784542`*^-16 y +
10 11
1.75857933767671115`*^-18 y - 1.93953952875018877`*^-21 y
12
+ 9.36627329152097587`*^-25 y
fZ = PolynomialFit[Zpoints, 12]
FittingPolynomial[<>, 12 ]
Expand[fZ[z]]
3.12900313166444732` + 0.0051494884384589028` z -
2 3
0.00465162337670335634` z + 0.000353810126643260947` z -
4 5
0.0000123168384545076659` z + 2.33973233023264981`*^-7 z -
6 7
2.68053725558395639`*^-9 z + 1.95944454066824906`*^-11 z -
8 9
9.36908222233828169`*^-14 z + 2.92259791649793854`*^-16 z -
10 11
5.7375125412051755`*^-19 z + 6.44124743834485258`*^-22 z -
12
3.15588230785927237`*^-25 z
ParametricPlot3D[{fX[t], fY[t], fZ[t]}, {t, 1, 339},
ViewPoint -> {1.30000000000000004`, 2.39999999999999991`, 2.`}]
![[Graphics:Bernoulli_Approxgr3.gif]](Bernoulli_Approxgr3.gif)
- Graphics3D -
The initial dataset of 339 points on the planar curve through the surface of the figure has been converted into a three-dimensional 12th-degree parametric equation. The higher the degree of the polynomial, the closer it approximates the original curve.
Compare this with Bernoulli's Limniscate [in Mathematica], which Bernard Morin theorized should approximate the same curve:
(x^2 + y^2)^2 - a^2|x^2-y^2| == 0
r^2 == a^2 Cos[2 \[Theta]]
(From: "CRC Handbook of Mathematical Curves and Surfaces", David H. von Seggern, Boca Raton: 1990, CRC Press, Inc., ISBN 0-8493-0155-6)
r := Sqrt[ a^2 Cos[2 \[Theta]]] /. a -> 1
ParametricPlot[{Re[Z], Im[Z]} /. Z -> r Exp[I \[Theta]], {\[Theta], 0, 2 Pi}]
Note that this, simpler representation, captures the abstract essence of the form, while discarding the eccentricities, in which the character of the figure may be encoded.
What other means do we have by which to intuitively interpret the qualities of
pure, physical form? Classes of mathematical forms such as
Minimal surfaces are characterized by their principal curvatures. Here
are the principal curvatures at the waist of our figure:
I propose a sculpture, in homage to Professor Bernard Morin. The figure will be reproduced life-sized in bronze or resin, with raised lines traced onto its surface, such as the lines of intersection depicted above. Traced along each line will be raised text, in DotsPlus Braille, specifying the mathematics of the curve being traced. Computer-aided design and manufacturing methods will be used to create this object.
In a similar fashion to the above, rather than fitting an equation to a single curve, one can fit a function in two parameters (such as a NURBS patch) to a parametrically 'rectangular' region on the surface of the figure scan model. By this method, one might begin to derive the equation of Bernard Morin's hypothetical 'specialized saddle shape'.
I do not at this moment have at my disposal the computer graphic tools necessary to carry out the above proposed two-parameter surface fitting. I propose to develop these, however.
Note that the expression for a single curve is quite long. To compose several of these on a single figure sculpture will be a challenge.
To do the same for a series of two-parameter functions will be even more difficult. I am guessing that the Functional Representation tools (HyperFun) may also be useful for this work.
Naum Gabo deconstructed the human figure to a pure essence of geometric forms. This was done in a very intuitive way. We now have the tools to do this directly, analytically and with full access to the language which is processed throughout the manipulation.
I propose to construct a figure sculpture by gleaning the pure forms from the digitization of a life model, via the methods described above. The forms composing the sculpture will be annotated in raised text ( Braille), relating the mathematics of the forms.
The generic human form, exemplified in the archaic Greek Kouros figure, and abstracted to the extreme in the constructivist manner above, has no character.
I propose that the Classical Greek figure can be constructed mathematically by a superposition of the functions describing the generic figure, overlaid with 'tuned' parameters which describe the unique character of an individual. I believe that this parametric system can be constructed using the techniques described above, in conjunction with other Functional Representation systems, such as HyperFun.
The mathematical model presented above is specific to this particular figure in this particular pose. More interesting, perhaps, might be a study of the aspects which make this figure unique with respect to all others.
An accurate anatomical model of a generic human figure can be posed within a commercially-available character animation system. This is now routinely done to animate synthetic characters in the motion picture industry.
I propose that these techniques be used to create physical sculpture with Braille or other raised-text captions which speak about the psychological state of the subject.