A lattice in S2×E.
Image by Coulon, Matsumoto, Segerman & Trettel

It is impossible to “find” a mathematical idea without explaining it; exploration and exposition are two sides of the same coin. One striking example of this is the epochal work of William Thurston; often his theorems were accompanied by pictures and computer programs, illustrating the underlying ideas. 

The Illustrating Mathematics Summer 2021 program (July 19-23) at the Park City Mathematics Institute (PCMI) will bring together mathematicians from a range of fields, and practitioners from the digital arts (animation, 3D printing, laser cutting, CNC routing, virtual reality, computer games, etc) to share their expertise in mathematics and with the procedural tools used to illustrate mathematics. Although officially a PCMI Graduate Summer School, the program is targeted both to graduate students and to mathematics community members from all backgrounds. In addition to online lectures from a variety of mathematicians and practitioners, our school will consist of several workshops to train participants in a variety of digital media.

To apply, follow PCMI’s published guidelines, but note that you are not required to be a graduate student (participants both without and with graduate education are welcome as well); the letter of recommendation is optional.

Program Details

Featured Plenary Speakers


(All times are U.S. Mountain Daylight Time.)

 Mon, Jul 19Tue, Jul 20Wed, Jul 21Thu, Jul 22Fri, Jul 23
8:30 - 9:00Come Say Hello:
informal tech check-in
9:00 - 9:25Welcome and
plenary meeting
9:30 - 11:00
Workshops (main sessions)
11:15 - 12:30
Workshop "virtual office hours"
13:00 - 14:00Daniel Piker
Plenary Talk
Vernelle Noel
Plenary Talk
Modular Origami
a participatory distributed
mathematical construction
Ingrid Daubechies
Plenary Talk
Workshops Present!
14:15 - 15:45Show and AskShow and AskCommunity
Show and Ask
16:00 - 17:30

Tech check-in

When convenient during this first half-hour on Monday morning, participants are encouraged to connect to the conference platform and informally say hello to the organizers. This period also provides an opportunity to ensure that the technology is working for you before the events begin.


All participants will have the opportunity to participate in one of the program’s workshops listed below. Participant/workshop matching will occur in advance (although participants may switch between Blender/Grasshopper or between CAD/CNC Toolpaths after the first joint session of these paired workshops). There will be an opportunity for participants to present what they’ve created on Friday afternoon.

Very brief summaries of the workshops are below; see the workshop page for much more detailed information.

The workshops will meet daily during the program. They will consist of a 90-minute primary presentation/work session supervised by the workshop leader, followed (after a 15-minute break) by “virtual office hours” with an experienced facilitator (the workshop leader or an assistant) for optional informal small-group discussion. Note there will also be virtual spaces set aside during peer collaboration time for participants to work together on workshop projects.

Blender for mathematicians: a beginner’s workshop
Rémi Coulon

Blender is an open source 3D graphics software, popular among digital artists for its organic virtual sculpting environment and advanced animation tools. This workshop will introduce beginners to its use for creating mathematical images.

Illustrating Mathematics with Grasshopper
Dave Bachman

Grasshopper is a visual scripting platform for Rhino 3D. Participants in this workshop will learn the basics of mathematical model building with Grasshopper.

Traditional CAD/CAM Software for Mathematical Models
Glen Whitney

This tutorial provides an introduction to traditional CAD/CAM drawing tools, a class of software that may be unfamiliar to many in the mathematics and mathematical illustration community.

Mathematical CNC Toolpaths
Edmund Harriss

CNC (Computer Numerical Control) machining has a rather high barrier to entry, but has a lot of mathematical potential. Participants will get to create lines and paths and see the effects when these get cut into wood.

Drawing Julia Sets
Arnaud Chéritat

We will review techniques for drawing Julia sets, dating from the 80s through recently developed tricks.

Graphic Novels Can Teach Mathematics
Audrey Nasar

In the United States, graphic novels are increasingly being used for scholastic purposes across the curriculum as supplements or replacements for traditional textbooks. In this workshop, participants will explore using this format for lessons of their own devising.

Visualizing Mathematical Structures with Processing
Roger Antonsen

We explore how mathematical structures, in particular discrete structures and sets of finite objects, can be visualized, and interactively experimented with, in the Processing graphics environment.

Interactive Videos for Teaching and Visualization
Yuri Sulyma

Participants will learn to create videos that viewers can manipulate/interact with, with a focus on mathematical content.

Show and Ask

In the spirit of what’s known in American culture as “Show and Tell,” these sessions will allow many participants a few minutes to say a bit about themselves and what their interests in illustrating mathematics are. Each participant is encouraged to display and describe a math-related image, object, video, audio recording, or other form of illustration, particularly one that is a work in progress, and ask the collected participants something they may be wondering about the illustration: how to improve it, how to solve some problem related to it, how to connect with others who might be interested, etc.

The Monday and Tuesday sessions will be reserved for registered participants in the Illustrating Math week and will help attendees get to know each other and become familiar with others’ efforts in this area. There will be an opportunity to sign up in advance, and any participant who wishes to display something and say a few words about it will have an opportunity. During the Thursday session, we will also welcome some members of the broader Illustrating Mathematics community to present, so that participants can become more familiar with the
range of efforts and inquiry going on in this field. If there is time available in any session once the scheduled presenters have “shown and asked,” the microphone will be opened to anyone else attending who may have something they would like to show and ask about.

Peer collaboration time

Time is set aside in the schedule each day for peer collaboration. During this time, there will be virtual spaces set aside dedicated to each workshop (without the presence of the workshop leader or assistants), as well as for general discussion on illustrating math topics. Participants may use this time to work together on workshop projects, initiate their own efforts to illustrate math together, or simply get to know each other better