Project Proposal:
Braille-Annotated Tactile Models In-The-Round
of Three-Dimensional Mathematical Figures

23115 Bluebird Drive Calabasas, CA 91302 USA

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.

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.

Keywords: Mathematics Education Blind Braille Access Three-Dimensional Model

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.

Color computer rendering of Costa's Genus 1 minimal surface

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).

Computer rendering of a Hyperbolic Paraboloid with text applied as
 flat surface maps

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.

Hyperbolic Paraboloid rendered in Stereolithography with captions
 printed on paper applied to the surfaces Hyperbolic Paraboloid rendered in Stereolithography with captions
 printed on paper applied to the surfaces Hyperbolic Paraboloid rendered in Stereolithography with captions
 printed on paper applied to the surfaces

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.

Hyperbolic Paraboloid rendered in Stereolithography with captions
 embossed in plastic sheet applied to the surfaces 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.


  1. T.A. DeFanti and M.D. Brown, "Visualization in Scientific Computing" (chapter), Advances in Computers, Vol. 33, Academic Press, pp. 247-305, Spring, 1991.

  2. David Hoffman, ; "New Embedded Minimal Surfaces", The Mathematical Intelligencer , Vol. 9, No. 3 (1987).

  3. Ivars Peterson, "Plastic Math", Science News, Vol. 140, No. 5, pp. 65-80 (August 3, 1991).

  4. Stewart Dickson, "Computer-Aided Rapid Mechanical Prototyping or Automated Fabrication" ;

  5. Stephen Wolfram, "Mathematica, A System for doing Mathematics by Computer", The Mathematica Book, Third Edition, Wolfram Media, Inc. and Cambridge University Press, 1996, ISBN 0-521-58889-8;

  6. Steven Wilkinson, 3D Plots of Implicitly Defined Surfaces,

  7. Mark Preddy, John Gardner, Steve Sahyun, and Dave Skrivanek Dotsplus: How-To Make Tactile Figures And Tactile Formatted Math Proceedings of the 1997 CSUN Conference on Technology and Persons with Disabilities, Los Angeles, CA, March 1997.