Steve Trettel

Steve Trettel

Hello! I'm a mathematician and artist at the University of San Francisco, working in geometry and mathematical visualization. This site collects my illustration, code, and writing. Sometimes I also take photographs and cook.

Art

Mathematical art that has been displayed in art shows, galleries, or museums.

Cubic Lines
Cubic Lines — Framed archival print
Paper Tori
Paper Tori — Framed Print
Elliptic Curves over Finite Fields
Elliptic Curves over Finite Fields — Gallery prints
Lightfall
Lightfall — Print on metal
Space(time)
Space(time) — Prints on metal (diptych)
Geodesic Boards
Geodesic Boards — Engraved wood (CNC)
Showing 6 of 18 artworks View full gallery →

Notes

Mostly notes to myself about things I'm trying to learn, or calculations from larger projects that may be interesting in their own right.

Showing 10 of 77 notes All notes →

Demos

Small web programs illustrating various mathematical objects, from both teaching and research.

O(2,2) on the Isotropic Grassmannian O(2,2) on the Isotropic Grassmannian O(4,2) on the Isotropic Grassmannian O(4,2) on the Isotropic Grassmannian Borsuk–Ulam Borsuk–Ulam Brouwer Fixed Point Brouwer Fixed Point Disk Map Playground Disk Map Playground The Hairy Ball Theorem The Hairy Ball Theorem Hypergeometric Atlas, Degree ≤ 6 Hypergeometric Atlas, Degree ≤ 6 Hypergeometric Atlas, Degree 7 Hypergeometric Atlas, Degree 7
Showing 8 of 417 demos Browse all →

Software

Larger systems and one-off tools, built mostly for research and teaching.

# constant-curvature-surfaces TypeScript with Stepan Paul Surfaces of revolution of constant curvature, textured by regions of the sphere or hyperbolic plane. # coxeter-viz TypeScript / three.js with Gordon Kirby Visualizing convex polytopes, Coxeter groups, and tilings in hyperbolic 2- and 3-space. # cubic-lines Javascript / GLSL with Claudio Gomez-Gonzales, Gabriel Dorfsman-Hopkins Real time visualizations cubic surfaces parameterized by six points in the projective plane. # elliptic-curve-viz TypeScript / GLSL with Nadir Hajouji Visualizing elliptic curves — over finite fields as point sets on flat tori in the 3-sphere, and over ℂ as complex tori in CP² — live in WebGL and path-traced for final renders. # geodesic-boards Javascript with Edmund Harriss Computing geodesics on surfaces z=f(x,y) and exporting for CNC cutting. # homogeneous-spaces TypeScript / three.js Drawing geometrically-correct points, geodesics, curves, and surfaces in 2D and 3D homogeneous spaces — constant-curvature geometries, their Lorentzian cousins, and product spaces — through swappable coordinate models. # knitted-surfaces JavaScript with MPI Math Lab Computing periodic knitted designs on surfaces. # lightfall TypeScript / three.js Rendering light cones in spacetimes warped by multiple black holes. # limit-sets TypeScript / three.js Rendering the limit sets of matrix groups acting on projective space.
Showing 9 of 19 software projects Details & documentation →

Graphics Archive

An archive of mathematical illustrations I've made over the years.

Limit Set Sp6 First Light Hyperbolic Disk in Euclidean Space The Hex Torus in S³ Airplug Hyperbolic Cube Analog Gravity Gravitational Portrait
Showing 8 of 291 graphics Browse all →

Writing

Long-form expositions, mostly notes from courses I've taught.

Modern Geometry

Course Notes 28 chapters · 90,000 words · 430 pages · 369 figures ( 22 interactive )

Notes / textbook for a 1 semester undergraduate course in geometry, taking calculus as the foundations. Covers Euclidean, Spherical and a bit of Hyperbolic geometry.

Multivariable Calculus

Course Notes 27 chapters · 35,000 words · 136 interactive figures

Supplementary Course Notes for the multivariable calculus course I teach at USF, with live animations.

Reaching for Infinity

Textbook 32 chapters · 71,000 words · 382 pages · 18 figures

A year-long advanced undergraduate real analysis textbook, with an emphasis on historically important problems.

Shader Programming

Course Notes 18 chapters · 57,000 words · 179 interactive figures

Course notes for GPU shader programming as a tool for mathematical visualization, covering fractals, implicit surfaces, tilings, and physical simulations in GLSL.

4 works All writing →

Academic

Papers, talks, exhibitions and project sites.

Fall 2026 previous semesters
  • Calculus I MATH 109 2 sections
  • Differential Equations MATH 340 1 section

Research PapersAll 13 →

Illustrating Hyperbolic Surfaces with Mesh Embeddings Bridges 2026, 2026 with Anna Wienhard, Diaaeldin Taha, Erik Loffelholtz, Fabian Lander
Classical Optics for Charged Black Holes Bridges Math Art 2025 Conference Proceedings, 2025
Elliptic Curves and the Hopf Fibration Bridges Math Art Conference Proceedings, 2025, 2025 with Nadir Hajouji

Art ExhibitionsAll 19 →

Elliptic Curves International Congress of Mathematicians, Philadelphia, PA — July 2026 with Nadir Hajouji
Warped Realities International Congress of Mathematicians, Philadelphia, PA — July 2026 with Edmund Harriss, Henry Segerman, Robert Fathauer, Stepan Paul
Creation: Between Art and Mathematics Maison Poincaré, Paris, France — April–July 2026 with Claudio Gomez-Gonzales, Gabriel Dorfsman-Hopkins

TalksAll 25 →

Mathematics of Beautiful Graphics MAA MathFest, Boston — Invited Address, 2026
Surfaces of Revolution in Homogeneous Spaces Casa Matemática Oaxaca (BIRS), 2026
Visualizing Elliptic Curves over Finite Fields Institut Henri Poincare, 2026
A Gravitational Photograph ICERM — Brown University, 2025

WebsitesAll 4 →

Algebraic Starscapes ↗ algebraicstarscapes.com with Edmund Harriss, Kate Stange
Visualizing Elliptic Curves over Finite Fields ↗ elliptic-curves.art with Nadir Hajouji
Geodesic Boards ↗ stevejtrettel.site with Edmund Harriss

On the road

When not at USF, I'm occasionally on the road for math (where I've recently been). I particularly enjoy giving colloquia, undergraduate talks, and public lectures — if you'd like to have me visit, please get in touch.

Nothing on the calendar yet — check back soon.