Triple Point Twist
This is one of the most elaborate entries in the exhibit, not because of the complexity of the image but because of the linkages. "Triple-Point Twist" appeared in slightly rotated form on the cover of the Notices of the AMS, as well as on the cover of a statistics volume by Iversen and Gergen [Statistics: The Conceptual Approach]. As chapter headings in that latter volume there were fourteen views of this object rotating in three-space. These can be accessed in the virtual exhibit either as an MPEG movie or as a virtual reality VRML document, enabling the viewer of the electronic version to interact with the object in ways impossible for the gallery visitor.
Even more significant than watching the object rotate is to view and review an MPEG movie that makes the object unfold, changing one of the parameters that twist the "bamboo curtain" ruled surface so that it intersects itself, forming a triple point. The equations defining the surface have been studied extensively by David Mond and Washington Marar.
It is these enhancements that represent the true innovation in such a virtual gallery. The viewer who becomes fascinated by one or another of the aspects of an object can investigate them at different levels, depending on the background and interests of the individual. In particular, it is possible in some cases to view and manipulate phenomena that relate the particular object to a wider area of mathematics.
Even more significant than watching the object rotate is to view and review an MPEG movie that makes the object unfold, changing one of the parameters that twist the "bamboo curtain" ruled surface so that it intersects itself, forming a triple point. The equations defining the surface have been studied extensively by David Mond and Washington Marar.
It is these enhancements that represent the true innovation in such a virtual gallery. The viewer who becomes fascinated by one or another of the aspects of an object can investigate them at different levels, depending on the background and interests of the individual. In particular, it is possible in some cases to view and manipulate phenomena that relate the particular object to a wider area of mathematics.
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This movie shows the triple-point surface rotating in space. You can see the Triple-Point from various angles. Several views similar to these appeared as chapter headings in the book Statistics: The Conceptual Approach, by Iverson and Gergen.
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Here, the surface is pulled apart so that the triple point is removed. The parameterization of the surface has a constant c that varies from -1 to 1 in this movie. At c = 0, the singularity disappears and the surface becomes embedded.
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The triple-point surface is a ruled surface, meaning that is can be swept out by a line moving over time. In this movie, we see this line as it generates the surface. The line first sweeps out a portion of the surface, then seems to backtrack, causing the surface to intersect itself. Finally, it curves back on itself a third time, and passes through the curve of self-intersection again, forming the triple point.
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The Mathematics of the Triple Point Twist
The surface shown in "Triple-Point Twist" is one from a series of surfaces described by David Mond and Washington Marar in "Real map-germs with good perturbations" , where they analyze a number of germs of singularities of surfaces. This particular example is a ruled surface given by the equations
(x,y,z) = (u, v3 + cv, uv + v5 + cv3),
where c is a parameter that can be varied. For values of c greater than 0, the surface has no self-intersection, but for values of c less than 0, a triple point and two pinch points appear. For the image shown in the exhibit, c = -1, but an MPEG movie is available showing a series of different values of c as it varies from -1 to 1. For each fixed value of u, allowing v to vary produces a planar curve in the plane x = u. The rulings for the surfaces are the straight lines produced when v is held fixed and u is allowed to vary. One of the MPEG movies shows the surface being swept out by these ruling lines.
(x,y,z) = (u, v3 + cv, uv + v5 + cv3),
where c is a parameter that can be varied. For values of c greater than 0, the surface has no self-intersection, but for values of c less than 0, a triple point and two pinch points appear. For the image shown in the exhibit, c = -1, but an MPEG movie is available showing a series of different values of c as it varies from -1 to 1. For each fixed value of u, allowing v to vary produces a planar curve in the plane x = u. The rulings for the surfaces are the straight lines produced when v is held fixed and u is allowed to vary. One of the MPEG movies shows the surface being swept out by these ruling lines.
Interactive version of the Triple Point Twist
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This is a VRML object that you can rotate and manipulate yourself. You can investigate the triple point and the self-intersection using this file.
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Triple Point Twist bibliography
- Washington Luiz Marar and David Mond, "Real map-germs with good perturbations", Topology 35 (1996), no. 1, 157--165.
- David Mond, "How good are real pictures?", Algebraic geometry and singularities (La Rábida, 1991), 259--276, Progr. Math. 134, Birkhäuser, Basel, 1996.
- David Mond, "Singularities of mappings from surfaces to 3-space", Singularity theory (Trieste, 1991), 509--526, World Sci. Publishing, River Edge, NJ (1995).
- David Mond, "Some remarks on the geometry and classification of germs of maps from surfaces to 3-space", Topology 26 (1987), no. 3, 361--383.