While the most efficient method for communicating math concepts is the use of real world objects or virtual manipulatives, creating illustrations of mathematical phenomena beyond three dimensions has been a particularly challenging task and many mathematical phenomena have thus only existed in the mathematician's mind. This research seeks to investigate the question of whether computer graphics techniques can help expert mathematicians and general public to visualize and communicate the higher-dimensional mathematical objects and their deformations. The ultimate goal of this research is to establish a mechanism by which an expert human viewer can manipulate a higher-dimensional geometric object that they can only see in part, i.e., via a slice or projection into two or three dimensions. Theoretical contributions of this research will impact and improve methods in mathematical visualization, particularly graph visualization, computer aided design, and large-scale spatial visualization, while the project deliverables will have direct and transformative impact on the ability of mathematicians to study higher-dimensional objects, and to communicate what they have learned in person, in presentations, and in archival works. The success of this project will ultimately translate into more rapid advancement in areas of pure and applied mathematics where higher dimensional geometry plays an important role, and provide mathematical visualization tool sets to facilitate college and K-12 students in their geometry courses. The project will make research outcomes including open source software freely available, and will disseminate mathematical sciences to the general public by rendering and presenting pedagogical animations at the University of Louisville Planetarium. The project also includes integrated educational and outreach activities for K-12, undergraduate, and graduate students.

The research will explore an interactive visualization paradigm that makes use of energy-driven self-deformable object models embedded in higher dimensions, supplemented by reduced-dimensional analogies for expert human viewers to guide the deformations towards their final goals. The investigators will begin by assigning a deformation energy to the higher-dimensional object, so that the aspects of the configuration that are unseen and unfamiliar can be controlled in a principled and well-posed manner by constraints or manipulations on the aspects of the configuration that are seen and familiar in our dimensions. Often times mathematical simulations are concerned with heavily vectorized operations performed over and over in a large number of iterations. The project will exploit hardware-enabled parallelism to accelerate mathematical simulations, and to extract key moments where successive terms differ by one critical change to represent and analyze various mathematical evolutions. By combining guided relaxation method and accelerated computation, this research can potentially make a novel contribution to building intuition about classes of geometric and topological problems that otherwise would be nearly impossible to communicate, perceptualize, and disseminate; and can potentially further the entire concept of the assistance and empowerment of human understanding by computer methods, specifically via the power of visual and computational spatial visualization tools. All outcomes of this project, including technical reports, research articles, links to educational and outreach activities, open source software, and pedagogical animations will be accessible from the project's web site (http://www.cecsresearch.org/vcl/nsf1651581/).

**J. Lin & H. Zhang. Accelerating visual communication of mathematical knot deformation. Journal of Visualization (2020).****H. Zhang & H. Liu. Relaxing topological surfaces in four dimensions. The Visual Computer volume 36, pages2341–2353(2020).****H. Liu & H. Zhang. A Suggestive Interface for Untangling Mathematical Knots. IEEE Transactions on Visualization and Computer Graphics(2020).**

*Abstract*: Mathematical knots are different from everyday ropes, in that they are infinitely stretchy and flexible when being deformed into their ambient isotopic. For this reason, challenges of visualization and computation arise when communicating mathematical knot’s static and changing structures during its topological deformation. In this paper, we focus on visual and computational methods to facilitate the communication of mathematical knot’ dynamics by simulating the topological deformation and capturing the critical changes during the entire simulation. To improve our visual experience, we design and exploit parallel functional units to accelerate both topological refinements in simulation phase and view selection in presentation phase. To further allow a real-time keyframe-based communication of knot deformation, we propose a fast and adaptive method to extract key moments where only critical changes occur to represent and summarize the long deformation sequence in real-time fashion. We conduct performance study and present the efficacy and efficiency of our methods.

*Abstract*: In this paper, we show the use of visualization and topological relaxation methods to analyze and understand the underlying structure of mathematical surfaces embedded in 4D. When projected from 4D to 3D space, mathematical surfaces often twist, turn, and fold back on themselves, leaving their underlying structures behind their 3D figures. Our approach combines computer graphics, relaxation algorithm, and simulation to facilitate the modeling and depiction of 4D surfaces, and their deformation toward the simplified representations. For our principal test case of surfaces in 4D, this for the first time permits us to visualize a set of well-known topological phenomena beyond 3D that otherwise could only exist in the mathematician’s mind. Understanding a fairly long mathematical deformation sequence can be aided by visual analysis and comparison over the identified “key moments” where only critical changes occur in the sequence. Our interface is designed to summarize the deformation sequence with a significantly reduced number of visual frames. All these combine to allow a much cleaner exploratory interface for us to analyze and study mathematical surfaces and their deformation in topological space.

*Abstract*: In this paper we present a user-friendly sketching-based suggestive interface for untangling mathematical knots with complicated structures. Rather than treating mathematical knots as if they were 3D ropes, our interface is designed to assist the user to interact with knots with the right sequence of mathematically legal moves. Our knot interface allows one to sketch and untangle knots by proposing the Reidemeister moves, and can guide the user to untangle mathematical knots to the fewest possible number of crossings by suggesting the moves needed. The system highlights parts of the knot where the Reidemeister moves are applicable, suggests the possible moves, and constrains the user's drawing to legal moves only. This ongoing suggestion is based on a Reidemeister move analyzer, that reads the evolving knot in its Gauss code and predicts the needed Reidemeister moves towards the fewest possible number of crossings. For our principal test case of mathematical knot diagrams, this for the first time permits us to visualize, analyze, and deform them in a mathematical visual interface. In addition, understanding of a fairly long mathematical deformation sequence in our interface can be aided by visual analysis and comparison over the identified "key moments" where only critical changes occur in the sequence. Our knot interface allows users to track and trace mathematical knot deformation with a significantly reduced number of visual frames containing only the Reidemeister moves being applied. All these combine to allow a much cleaner exploratory interface for us to analyze and study mathematical knots and their dynamics in topological space.

**J. Lin & H. Zhang. Visualizing Mathematical Knot Equivalence. Electronic Imaging, Visualization and Data Analysis 2019, pp. 683-1-683-11(11).****J. Lin & H. Zhang. Visually Communicating Mathematical Knot Deformatione. VINCI'2019: Proceedings of the 12th International Symposium on Visual Information Communication and Interaction. September 2019 Article No.: 20 Pages 1-8.****J. Lin & H. Zhang. View Selection in Knot Deformatione. 2019 IEEE International Conference on Big Data (Big Data), Los Angeles, CA, USA, 2019, pp. 3365-3372.****G. Ruan & H. Zhang. Parallelized Topological Relaxation Algorithm. 2019 IEEE International Conference on Big Data (Big Data), Los Angeles, CA, USA, 2019, pp. 3406-3415.**

*Abstract*: We present a computer interface to visualize and interactwith mathematical knots, i.e., the embeddings of closed circles in3-dimensional Euclidean space. Mathematical knots are slight-ly different than everyday knots in that they are infinitely stretchyand flexible when being deformed into their topological equiva-lence. In this work, we design a visualization interface to depict mathematical knots as closed node-link diagrams with energiescharged at each node, so that highly-tangled knots can evolve by themselves from high-energy states to minimal (or lower) energystates. With a family of interactive methods and supplementary user interface elements, our tool allows one to sketch, edit, andexperiment with mathematical knots, and observe their topological evolution towards optimal embeddings. In addition, our inter-face can extract from the entire knot evolution those key momentswhere successive terms in the sequence differ by critical change;this provides a clear and intuitive way to understand and trace mathematical evolution with a minimal number of visual frames.Finally our interface is adapted and extended to support the depiction of mathematical links and braids, whose mathematicalconcepts and interactions are just similar to our intuition aboutknots. All these combine to show a mathematically rich interfaceto help us explore and understand a family of fundamental geometric and topological problems.

*Abstract*: Mathematical knots are different from everyday ropes in that they are infinitely stretchy and flexible when being deformed into their ambient isotopic. For this reason, a number of challenges arise when visualizing mathematical knot's static and changing structures during topological deformation. In this paper we focus on computational methods to visually communicate the mathematical knot's dynamics by computationally simulating the topological deformation and capturing the critical changes during the entire simulation. To further improve our visual experience, we propose a fast and adaptive method to extract key moments where only critical changes occur to represent and summarize the long deformation sequence. We conduct evaluation study to showcase the efficacy and efficiency of our methods.

*Abstract*: Extracting good views from a large sequence of visual frames is quite difficult but a very important task across many fields. Fully automatic view selection suffers from high data redundancy and heavy computational cost, thus fails to provide a fast and intuitive visualization. In this paper we address the automatic viewpoint selection problem in the context of 3D knot deformation. After describing viewpoint selection criteria, we detail a brute-force algorithm with a minimal distance alignment method in a way to not only ensure the global best viewpoint but also present a sequence of visually continuous frames. Due to the intensive computation, we implement an efficient extraction method through parallelization. Moreover, we propose a fast and adaptive method to retrieve best viewpoints in real-time. Despite its local searching nature, it is able to generate a set of visually continuous key frames with an interactive rate. All these combine provide insights into 3D knot deformation where the critical changes of the deformation are fully represented.

*Abstract*: Geometric problems of interest to mathematical visualization applications involve changing structures, such as the moves that transform one knot into an equivalent knot. In this paper, we describe mathematical entities (curves and surfaces) as link-node graphs, and make use of energy-driven relaxation algorithms to optimize their geometric shapes by moving knots and surfaces to their simplified equivalence. Furthermore, we design and conFigure parallel functional units in the relaxation algorithms to accelerate the computation these mathematical deformations require. Results show that we can achieve significant performance optimization via the proposed threading model and level of parallelization.

**J. Lin & H. Zhang. Untangling Mathematical Knots with Simulated Annealing and Opposition-Based Learning. 2018 IEEE International Conference on Big Data (Big Data), 4245-4253.****H. Zhang. Performance Engineering for Scientific Computing in R. International Journal on Data Science and Technology 4(2), 42-48.**

*Abstract*: The fundamental problem in knot theory is to determine whether a given knot is an unknot. Solving such problemwith existing computational algorithms or heuristic methods still remains challenging. In this paper we describe a hybrid algorithmthat employs simulated annealing (SA) algorithm and opposition-based learning strategy to simplify and accelerate the simulationof mathematical knot relaxation. In our work, mathematical knots are modeled as closed node-link diagrams with energiescharged at each node. A mathematical force model with adaptivestep sizes is incorporated into SA for new individual generation. One elementary move with a descending number of nodes isincluded to prune the conformation and reduce the redundant information following the isotopy consistency. However, the force-directed model may be trapped to a local energy minimum. Toovercome such intrinsic limitation, we adopt an opposition forcebased learning strategy to tackle this problem. With a flexibleneighborhood and the instructive disturbance information, theknot geometry is able to untangle quickly and can reach the global minimum state without extra intervention. Illustrative testcases are presented, including three unknots and two classicalknots with complex isotopy.

*Abstract*: R has been adopted as a popular data analysis and mining tool in many domain fields over the past decade. As Big
Data overwhelms those fields, the computational needs and workload of existing R solutions increases significantly. With
recent hardware and software developments, it is possible to enable massive parallelism with existing R solutions with little to
no modification. In this paper, three different approaches are evaluated to speed up R computations with the utilization of the
multiple cores, the Intel Xeon Phi SE10P Co-processor, and the general purpose graphic processing unit (GPGPU).
Performance engineering and evaluation efforts in this study are based on a popular R benchmark script. The paper presents
preliminary results on running R-benchmark with the above packages and hardware technology combinations.

**R. Chen & H. Zhang. Large-scale 3D Reconstruction with an R-based Analysis Workflow. In Proceedings of the Fourth IEEE/ACM International Conference on Big Data Computing, Applications and Technologies, 85-93.**

*Abstract*: As the volume of data and technical complexity of large-scale analysis increases, many domain experts
can no longer be seated in the data exploration and analysis workflow. What is desired is a computational
powerful but still familiar analysis interface for domain experts to fully participate in the analysis workflow
by just focusing on individual datasets, leaving the large-scale computation to the system. Towards this goal,
we present VisRden, a research prototype that combines user friendly visual programming and scalable computing backend for
large-scale 3D reconstruction in carious lesion research. VisRden uses R as the analysis language, making a set of core functions
available to the users by hiding the computational complexity behind a visual interface, and allowing advanced users to provide custom
R scripts and variables to be fully embedded into the final analysis script. Using R as the analysis language allows cariologists to continue
explore data and propose new analysis methods in the way they are already familiar with. VisRden conquers large-scale image processing and 3D
reconstruction in a MapReduce-like framework using R and SGE (Sun Grid Engine) array jobs. Image-based operations and result aggregation are
scheduled as array jobs in a parallel means to accelerate the knowledge discovery process. All these combine to provide a new analytics workflow
for performing similar large-scale analysis loops that need expert users to closely supervise, provide feedback, and refine the subtasks.

**J., Lin, D. Zhong, Y. Zhong, & H. Zhang. Accelerating mathematical knot simulations with R on the web. 2016 IEEE International Conference on Big Data (Big Data), 2315-2321.****H. Zhang, Y. Zhong, J. Jiang. Visualizing Knots and Braids with Touchable 3D Manipulatives. 2016 IEEE Pacific Visualization Symposium (PacificVis), 24-31.**

*Abstract*: Many geometric problems of interest to mathematical visualization applications involve
changing structures requiring mathematical simulations, such as the moves that transform one knot into an equivalent knot.
In this paper, we propose to explore a unique paradigm that makes use of self-deformable object models embedded in mathematical
space, supplemented by energy-driven relaxation and user-defined motion constraints to guide the simulation through the configuration space.
Furthermore, we exploit web-based user interfaces and parallelization to accelerate mathematical simulations, and to extract key moments where
successive terms in the sequence differ by one critical change to represent and analyze various mathematical evolutions. The presented work leverages
the nature of the interrelationship between mathematics and computer science, especially computer graphics, graph algorithms, user interfaces,
and accelerated computation.

*Abstract*: In this paper we present a mathematical knot/braid diagram interface
that exploits 3D computer graphics, interactive visualization, and multi-touch technology to
enhance one's intuitive experience with mathematical theory of knots. Our interaction model is based
on the clever but simple geometric construction named the Reidemeister moves, which allows 3D topological
manipulations using rather simple 2D moves. Multi-touch interfaces can provide a natural way for us to interact
with the extra degrees of freedom that characterize knots' mathematical, physical, and arithmetic properties. Relative
to a specialized mouse-driven interface, the proposed multi-touch interface is easier and more intuitive to learn, and our
pilot study shows that knot and braid manipulation with multi-touch is much faster and more efficient. All these combine to
show that interactive computer graphics methods and computer interfaces can be used to construct virtual manipulatives and
meet the challenge of exploring abstract mathematical worlds.

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**Juan Lin**, *Fall 2016 - now*

**Huan Liu**, *Fall 2018 - now*

**Summer 2018:** Together with other CECS labs, we hosted high school STEM teachers in a 6-week RET program for
the STEM teachers to participate in our research activities with faculty and phd students in labs. Project: learning and using R for scientific visualization.

**Summer 2019:** Together with other CECS labs, we hosted high school STEM teachers in a 6-week RET program for
the STEM teachers to participate in our research activities with faculty and phd students in labs. Project: mathematical knot creation from hand-drawn images.

Last updated on March 15, 2020.