Fig. 1 --
Costa's Minimal Surface
Image by David Hoffman and James Hoffman
STEWART DICKSON
http://emsh.calarts.edu/~mathart
Making images of multi-dimensional computer data transforms the information into a form accessible by a different sensory modality, opening the problem to interpretation from a differing point-of-view. This often leads to unexpected discoveries in a system which was previously only accessible through a highly abstract means. The author has demonstrated this transformation into three physical dimensions using Rapid Prototyping Technologies.Keywords: Mathematics Education Blind Braille Access Three-Dimensional Model
A computer model can further parametrically map an additional dimension of information onto the surface of a three-dimensional object, as a surface texture map, for example.
In this paper, the author demonstrates a technique of applying captions in Braille to the surface of a three-dimensional, in-the round mathematical figure in a computer modeling system. The author terms this operation a higher-dimensional integration of information. This technique can provide access to understanding of three-dimensional mathematics for the visually impaired.
1. Background and Motivation
Visualization in Scientific Computing has brought about historical
breakthroughs or "quantum leaps" in understanding by transforming and
concretizing a highly abstract system. [1]
One such breakthrough was the Costa-Hoffman-Meeks family of minimal surfaces,
discovered in 1983.
[2]
By the 1890's the study of minimal surfaces was thought to be exhausted -- no
new surfaces could be described mathematically which were non-self-intersecting
(embedded) in three-space and which had vanishing mean curvature. However, in
1983 a graduate student in Rio de Janeiro named Celsoe Costa wrote down an
equation for what he thought might be a new minimal surface, but the equations
were so complex, they obscured the underlying geometry.
David Hoffman at the University of Massachusetts
at Amherst enlisted
James Hoffman
to make computer-generated pictures of
Costa's surface (Figure 1). The pictures they made suggested first, that the
surface was probably embedded - which gave them definite clues as to the
approach they should take toward proving this assertion mathematically - and
second, that the surface contained straight lines, hence symmetry by reflection
through the lines.
Fig. 1 --
Costa's Minimal Surface
Image by David Hoffman and James Hoffman
The symmetry led Hoffman and William Meeks, III to extrapolate that the surface was radially periodic and that new surfaces of the same class could be achieved by increasing the periodicity. They did so by altering the mathematical description of the surface to be the solution to a boundary-value problem constrained by the behavior of a minimal surface at the periodic lines of symmetry. The result: Hoffman and Meeks proved that Costa's surface was the first example of an infinitely large class of new minimal surfaces which are embedded in three-space.
The technique Hoffman and Meeks used was to make a picture which caused them to modify their mathematical theory and discover something totally unexpected about that theory. They later extended their techniques to find minimal surfaces of more complex geometry and they also created pictures of them. This is a new kind of experimental mathematics and a procedure not far from creative visual art.
The author has in turn taken some of these historical examples, visualized on the computer screen, and produced physical 3-D output from the computer data, using computer-aided rapid prototyping devices. [3] [4]
The power of visualization has been proven to a fairly convincing degree by the history cited here, but can similar arguments be made for tactilization? Clearly the segment of our population who cannot use computer pictures because their eyes do not work well enough to see the pictures have a real need for concrete artifacts from cyberspace.
The author argues further that the physical presence provides a degree of immediacy missing from computer graphic displays, which in turn supplies more information than a two-dimensional picture. However, for the human observer, the meaning of the mathematical language which originates the form can become disconnected from the form when it becomes physical. The author proposes that improvement can be made by mapping descriptive text onto the three-dimensional object (See Figure 2).
Fig. 2 -- Annotated Hyperbolic Paraboloid
Image by Stewart Dickson
2. Information Integration in Three-Space
In Figure 2,
a portion of a hyperbolic paraboloid has been mapped with text information
indicating special features of the surface -- in particular, the curves
which result when one of the parameters in the equation is held to a constant:
The Parabolas at X=0 and Y=0, the Hyperbolas parallel to the X-Y plane,
achieved by setting Z to a non-zero constant, and the degenerate hyperbola, or
two straight lines which lie in the X-Y plane at Z=0.
Figure 3 shows three views of a stereolithograph made from a Hyperbolic
Paraboloid computed in the Mathematica system for doing mathematics by
computer.[5] The surface was truncated at the
cube [-4,-4,-4],[4,4,4] using the ImplicitPlot3D Mathematica package by
Steven Wilkinson, Northern Kentucky University. [6]
The object occupies a cubic volume, eight inches on an edge. The surface
is one-eighth of an inch thick. Thickness was created from the theoretical
surface using software written in C++ by the author.
Fig. 3 -- Three Views of an Annotated Hyperbolic Paraboloid Stereolithograph
Dimensions: 8" X 8" X 8"
Images by Stewart Dickson
In this figure, the captions were printed in PostScript onto paper using a laser printer and then applied using adhesive to the surface of the model.
Further integration of information and access to the visually-impaired can be achieved by translating the text into Braille.
Figure 4 shows two views of a stereolithograph of the Hyperbolic Paraboloid with captions in Braille applied to it.
ALT="Hyperbolic Paraboloid rendered in Stereolithography with captions
embossed in plastic sheet applied to the surfaces"
Fig. 4 -- Two Views of Hyperbolic Paraboloid Stereolithograph Annotated in
DotsPlus Braille
Dimensions: 8" X 8" X 8"
Images by Stewart Dickson
The Braille font used in this figure was the DostPlus proposed standard for mathematical typesetting in Braille, developed by the Science Access Project, Department of Physics, Oregon State University. [7] The captions were printed onto self-adhesive plastic sheets using the TIGER 1000 personal tactile graphics embosser.
3. Conclusion and Further Development
Computer surface texture mapping is not new. Jim Hoffman actually used
it in 1985 to encode the Gaussian curvature of the surface
as a color map in the images he made of minimal surfaces, as in Figure 1.
The novel improvement shown in this paper is to apply text in Braille and
graphical features to the surface of a physical model, in-the-round of a
computer-generated three-dimensional object. In particular, the text and
graphics supply abstract information on the portion of the surface on which
the text lies.
The author believes that a corporeal synergy occurs when "reading" this
multi-dimensional, tactile, self-describing, integrated information object.
As the reader's hand passes over the text, describing the surface at that
region, the hand is constrained to the surface under all six degrees of
freedom -- that is, the hand not only occupies a three-space position
determined by the mathematics of the surface, but it is also oriented
according to the normal vector to the surface and the tangent vector along
which the text lies in the surface.
The author believe that this corporeal awareness of the object is a synergetic
experience in which the effect is greater than the sum of the individual
elements.
However, this assertion needs to be tested, and the tactile exhibits shown
her are not quite ready for heavy use in a laboratory classroom.
The compound curvature of the surface in this demonstration made application
difficult of artwork printed onto two-dimensional stock. In addition, the
surface of the stereolithograph contains "steps" at 0.005-inch intervals,
resulting from the slice-wise method of construction used in computer-aided
rapid prototyping. These features tend to thwart adhesion of the artwork onto
the surface.
The author proposes that the object should be properly mapped with
dimensional Braille text geometry within the 3-D computer modeling system,
prior to three-dimsnsional output.
This works fairly well for easily parameterized surfaces. However in
general, output of mathematical graphs from Mathematica for example, contain
polygon geometry without any parametric orientation. Therefore a mapping
facility must be used which can parameterize a generic polygon mesh via a
geodesic path boundary, for example.
References