Corpus Mathematica

Public Domain Cyberware Scan r96033 OBJ file with Textured Squares containing the Seven Chakras -- Screen dumps from OpenGL/GLUT on MacOS-X 10.10 Yosemite

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 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

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.

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.

The curve, shown above in Open Inventor, is converted to data readable by Mathematica:

<< figure-1_bernoulli_geom.m ;

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