Scott B. Lindstrom

Gallery

At the core of my research is the construction of computational models. Whenever possible, I strive to include visualizations of them in my published work, because they illuminate the path of discovery for the reader. It is one thing to read a difficult proof; it is another to actually see how it was found. This page contains images that I find to be especially beautiful. This page is organized by the topic of research to which the images belong, and the topics featured are as follows:

Dynamical systems and splitting algorithms
Non-Euclidean Geometry
Analysis -- Functional and Complex
Conic programming, duality, and error bounds
Other machine learning topics
Just for Fun

Dynamical systems and splitting algorithms

This image is from the article The Douglas--Rachford Method for Ellipses and p-Spheres, a joint work with Jonathan M Borwein, Brailey Sims, Matthew P Skerritt, and Anna Schneider. This image shows how basins of periodicity may emerge for an iterative method applied to solving a perturbed version of the phase retrieval problem. This image was produced with the dynamical geometry scripting language Cindyscript.

Here we see the same feasibility problem through a different visualization method. Rather than showing individual iterates, we run the Douglas--Rachford algorithm starting from each individual pixel in the plane. We assign to each pixel a point, and then we color the pixel according to the behaviour of the point under the dynamical system. Colors were inspired by Australian aboriginal artwork. While the swirling patterns shown in the first image above might intuitively suggest curved basins, those which emerge in this image appear more polyhedral than one might expect.

Distinctions:
(1) This picture is the main poster image for Australian Mathematical Society special interest group Mathematics of Computation and Optimization (MoCaO). It, and its related variants, appears in all e-mails and on the website.

(2) This image was featured at the Bridges Math Art conference in Stockholm, Sweden.

Algorithms exhibit dynamics. Some are sprinters; others are dancers. A well-choreographed dancer explores the space in a way that points to the solution. If we learn the choreography, then we can monitor the dancer and use their steps to predict and test possible solutions on the fly. These on-the-fly predictions form a new algorithm, called a meta-algorithm. This image reveals the dynamics of such a meta-algorithm. It too appears to be dancing.

This image is from joint work with Alberto De Marchi. It is featured in the 2022 CARMA/AustMS Art Gallery.

Non-Euclidean Geometry

This is a rotation of hyperbolic space plotted on a pseudosphere. Each color represents a geodesic that is mapped under the transformation to another geodesic called a "tractrix" that runs up the side of the pseudosphere from its rim to its top. One tractrix is visible where the colors are cut. Black regions are mapped to parts of hyperbolic space that are outside of the region shown on the surface.

Distinctions: This image is part of a work, co-authored with Paul Vrbik, that published in Mathematical Intelligencer.

These images show symmetrical rotations on hyperbolic space. The intrinsic geometry of hyperbolic space is actualized in the extrinsic geometry of a pseudosphere. By slicing the pseudosphere and twisting it downwards, we obtain Dini's surface, a representation that allows us to see the pseudosphere wrapping around more than one time. The colors all meet together beyond the pseudosphere's rim at a "point at infinity." Unique colors highlight curves that are geodesics.

Distinctions: This image was voted most popular at A Night of MAPS (Mathematical and Physical Sciences) and Art at University of Newcastle in 2016.

Three rotations of hyperbolic space -- this rendering shows three different rotations of hyperbolic space. It appears in my book chapter in Handbook of the Mathematics of the Arts and Sciences.

Distinctions:

(1) First prize in the art category for the inaugural annual CARMA-MATRIX poster competition, outcome announced at the annual meeting of the Australian Mathematical Society.

(2) Judges' choice award in the same event.

(3) This image now also serves as the poster image for this recurring event.

Here we see two different translations of hyperbolic space plotted with different techniques on the Poincare disc. Each technique assigns unique colors to preimages of geodesics in the space.

Distinctions: These images are part of a work, co-authored with Paul Vrbik, that published in Mathematical Intelligencer.

Analysis -- Functional and Complex

This image is entitled Energy, Entropy, and Homotopy. It displays a continuous transformation of the energy function (dark blue) into the Boltzmann-Shannon entropy (dark red). The latter has as its domain the non-negative real numbers, and yet a continuous transformation (homotopy) is possible with respect to the epi toplogy. The method in question, the proximal average, makes use of the Lambert W function, a branch function, and was discovered through a combination of symbolic algebra and experimental computation. This image appears in a joint work with Heinz H. Bauschke in the Pure and Applied Functional Analysis special issue in memory of Jonathan Borwein.

I generated this image for my chapter in Handbook of the Mathematics of the Arts and Sciences, because it connects two themes of experimental mathematics that were emphasized by Borwein.

It is entitled: A very Riemannian situation: the function that bears his name plotted on the sphere that bears his name. Riemann conjectured his famous hypothesis after finding roots of his famous function by hand. Had he had access to modern computational tools, he might have saved some time. An unexceptional computer generated this 640,000 point image in a mere ten hours.

Distinctions: Judges' Choice award in the AustMS/CARMA mathematics/art poster competition in 2020.

As the energy is symmetric about zero, we can easily adapt the homotopy from Energy, Entropy, and Homotopy into a homotopy between the entropy and its own symmetric twin about zero. This image is titled Energy, Entropy, Homotopy, and Symmetry.

Conic programming, duality, and gauge optimization

This image is called The three-halves p-cone in four dimensions. It is composed of five separate images that all reveal the same object. That object is a cone in 4 dimensions. Just as an architect draws 2-dimensional cross sections of a 3-dimensional house, each image here shows a 3-dimensional cross section of the 4-dimensional cone. When you examine the work from the left, right, top, bottom, and center, you see the cone from different perspectives. When you stand back, the four images come together as one united idea.

This composite image is based on the joint work with Bruno Lourenco and TK Pong: Optimal error bounds in the absence of constraint qualifications with applications to the p-cones and beyond.

It is featured in the 2023 CARMA/AustMS Art Gallery. That year's judges provided the following artistic review:

An intriguing image. Engaging pastel colours juxtaposed with higher contrasted and saturated colours in the centre. Fantastic symmetry and balance.

A related image called Seeing in Higher Dimensions was featured in the Heidelberg Laureate Forum Intercultural Science-Art Project.

This image shows the exponential cone and its dual. Using parametric plotting and the Lambert W function, we color the cones according to the collections of faces that are of special interest for proving error bounds. It is from a joint project with Bruno Lourenco and TK Pong.

This image shows a lower dimensional slice of a graph of a function on a 3-manifold. The function represents a lower bound on a ratio of two monotonic functions composed on pairwise distances between points. By using computer algebra systems to examine slices like this one, we can visually discover when error bounds are Lipschitz, Hölderian, or even more exotic. These images reveal something even more interesting: a branch change in the Lambert W function, caused by set-valuedness of primages under the projection operator. This sneaky phenomenon was a barrier to a complete proof for many months. Once it was discovered, a clever change of variables---based on geometry---removed the obstruction. This image was created as part of an investigation with Bruno Lourenco and TK Pong.

A version of it appears in my book chapter, "The Art of Modern Homo Habilis Mathematicus, or: What Would Jon Borwein Do?" in Handbook of the Mathematics of the Arts and Sciences.

This image illustrates the construction of the projected polar proximal point algorithm and the polar envelope. It appears in my article, "The Projected Polar Proximal Point Algorithm Converges Globally."

Just for Fun

The above image shows me presenting a poster in virtual reality at the 8th Heidelberg Laureate Forum in September, 2021. The 8th HLF took place during the global coronavirus pandemic, and so was held online. In the foreground, taking the screen capture, is my friend, frequent co-author, and fellow HLF participant Neil Dizon, whom I had not seen in person since 2019. Such images are a glimpse into a moment in mathematics history. While mathematics has been with us through every catastrophic global pandemic, this was the first of the information age, and we adapted by finding new ways of being together. In the below image, Neil presents his poster.

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