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Prosection views: Dimensional inference through sections and projections
 Journal of Computational and Graphical Statistics
, 1994
"... We present some basic properties of two general graphical techniques for constructing views of highdimensional objects, projection and section. Projections can easily display aspects of structure that are only of low dimension, while sections, i.e., intersections of subspaces with a highdimensional ..."
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Cited by 43 (5 self)
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We present some basic properties of two general graphical techniques for constructing views of highdimensional objects, projection and section. Projections can easily display aspects of structure that are only of low dimension, while sections, i.e., intersections of subspaces with a highdimensional object, can easily display structure of only low codimension (and hence often high dimension). However, compositions of sections and projections, here called prosections, can display aspects of structure of any intermediate dimension. These statements are relevant for data analysis: projections of data can be easily generated with xyscatterplots, 3D data rotations, and grand tours, while sections can be approximated in existing systems by scatterplot brushing and painting. Thus this paper is in part an investigation into the principles underlying these techniques.
Computers, Reasoning and Mathematical Practice
"... ion in itself is not the goal: for Whitehead [117]"it is the large generalisation, limited by a happy particularity, which is the fruitful conception." As an example consider the theorem in ring theory, which states that if R is a ring, f(x) is a polynomial over R and f(r) = 0 for every e ..."
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Cited by 7 (3 self)
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ion in itself is not the goal: for Whitehead [117]"it is the large generalisation, limited by a happy particularity, which is the fruitful conception." As an example consider the theorem in ring theory, which states that if R is a ring, f(x) is a polynomial over R and f(r) = 0 for every element of r of R then R is commutative. Special cases of this, for example f(x) is x 2 \Gamma x or x 3 \Gamma x, can be given a first order proof in a few lines of symbol manipulation. The usual proof of the general result [20] (which takes a semester's postgraduate course to develop from scratch) is a corollary of other results: we prove that rings satisfying the condition are semisimple artinian, apply a theorem which shows that all such rings are matrix rings over division rings, and eventually obtain the result by showing that all finite division rings are fields, and hence commutative. This displays von Neumann's architectural qualities: it is "deep" in a way in which the symbol manipulati...
A Family of SinglyPeriodic Minimal Surfaces Invariant Under a Screw Motion
, 1993
"... This paper explains, among other things, how we were able to make the pictures of these surfaces, which appear at the beginning of Section 2. ..."
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Cited by 5 (0 self)
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This paper explains, among other things, how we were able to make the pictures of these surfaces, which appear at the beginning of Section 2.
Infinite periodic discrete minimal surfaces without selfintersections
 Balkan J. Geom. Appl
"... Abstract. A triangulated piecewiselinear minimal surface in Euclidean 3space R 3 defined using a variational characterization is critical for area amongst all continuous piecewiselinear variations with compact support that preserve the simplicial structure. We explicitly construct examples of suc ..."
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Abstract. A triangulated piecewiselinear minimal surface in Euclidean 3space R 3 defined using a variational characterization is critical for area amongst all continuous piecewiselinear variations with compact support that preserve the simplicial structure. We explicitly construct examples of such surfaces that are embedded and are periodic in three independent directions of R 3.
Through a Glass Darkly 1 Prolegomena
, 807
"... Education is a repetition of civilization in little. ..."
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Visualization in Learning Mathematics with
"... Visualization has always been an essential aid in the communication of mathematics. It is an important way to concretize concepts, to develop abstraction skills, and to motivate learning, for example in topology and geometry, and in the application of numerical methods to simulations of the real wor ..."
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Visualization has always been an essential aid in the communication of mathematics. It is an important way to concretize concepts, to develop abstraction skills, and to motivate learning, for example in topology and geometry, and in the application of numerical methods to simulations of the real world. Video has proven to be one of the most adequate ways to communicate visualization results, allowing to present in a rich cultural context a large quantity and diversity of information in a brief period of time. However, by itself, video has a limited capability to support learning. The structure and interaction introduced by hypervideo allow providing the user with greater control and autonomy, exploring links among the information conveyed by the video and complemented by other materials, augmenting its capabilities as a cognitive artifact. This paper develops these ideas, presenting The Story of Pi hypervideo as a case study.
Contents
, 2000
"... This formal development defines µJava, a small fragment of the programming language Java (with essentially just classes), together with a corresponding virtual machine, a specification of its bytecode verifier and a lightweight bytecode verifier. It is shown that µJava and the given specification of ..."
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This formal development defines µJava, a small fragment of the programming language Java (with essentially just classes), together with a corresponding virtual machine, a specification of its bytecode verifier and a lightweight bytecode verifier. It is shown that µJava and the given specification of the bytecode verifier are typesafe, and that the lightweight bytecode verifier is
Mathematical Visualization and Online Experiments
, 2000
"... The future of mathematical communication is strongly related with the internet. On a number of examples, the present paper gives a futuristic outlook how mathematical visualization imbedded in the internet will provide new insight into complex phenomena, influence the international cooperation of ..."
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The future of mathematical communication is strongly related with the internet. On a number of examples, the present paper gives a futuristic outlook how mathematical visualization imbedded in the internet will provide new insight into complex phenomena, influence the international cooperation of researchers, and allow to create online hyperbooks combining interactive experiments and mathematical texts.