Read the kind of claim
The labels in each exhibit describe what supports a statement. Its assumptions and references are part of the statement, and should travel with any conclusion drawn from it.
- Theorem
- An established mathematical result, stated with the conditions under which it applies.
- Derivation
- A sequence of mathematical steps explaining a formula or applying an established result.
- Numerical experiment
- A computation with recorded inputs, a method, actual outputs and numerical limitations.
- Visualization
- A view of an object or computed data. Colors, projections and animation help interpretation; the supporting argument remains in the text.
- Exploration question
- A prompt for further investigation whose answer has not been established by the question itself.
Finite precision and diagnostic values
Calculations use JavaScript’s binary64 numbers, commonly called double precision. Displayed decimal values are rounded for reading; subsequent calculations and JSON/CSV exports use the stored precision. A residual reports the computed difference in a named identity or equation. Its tolerance depends on the algorithm, input scale and problem size.
Small residuals support numerical consistency within the stated test. They do not establish exact arithmetic or a new theorem. Raw tiny negative eigenvalues or heat values can occur through roundoff and remain in Spectral exports with diagnostics. Color scales may use a limited display range; that choice does not alter the numerical values.
Compare independent quantities
Known-answer examples and independently computed quantities help expose mistakes. Fourier compares error measured in sample space with energy in omitted coefficients. Geometry compares spherical angle excess with a separate vector-based area expression. Spectral checks eigenpair residuals and orthogonality, and uses small graphs with analytic spectra and heat solutions.
A value derived from a formula must be identified as such. In particular, the displayed hyperbolic triangle area is calculated from angle defect; substituting that value into the same identity is not an independent test of the theorem. Chaos separates an initial-condition perturbation from a change in integration step, comparing numerical solutions only at common times.
Sources: [M2] Discrete Fourier Transform — NumPy reference · [M4] Geometry with an Introduction to Cosmic Topology — Section 5.4 · [M9] CSE 168 — Materials lecture · [M8] Spectral and Algebraic Graph Theory · [M7] Deterministic Nonperiodic Flow
Keep an experiment reproducible
An experiment record keeps its exhibit and model version, complete parameters, method, software identifier, actual creation time, measurements, warnings and references. Input contours and other required data travel in the JSON record. Recorded study batches also identify their execution environment and source version. A rerun compares numerical content while treating timestamps as run metadata.
Save snapshot stores an experiment in the current browser. Notebook lists those local records for reopening or deletion. JSON preserves the complete experiment; CSV provides its tabular values. A parameter link restores supported small inputs, while a custom drawing is reproduced with its JSON file. Export a file to retain a copy outside this browser profile.
Import checks the format version, required fields, parameter domains, array bounds and finite values before recomputing from the inputs. Imported content is data; it does not supply executable expressions. Errors describe what could not be read or computed. A threshold that was never observed remains unobserved within the stated window.
Sources, original work and review
The reference list identifies textbooks, author-created explanations, papers and official technical documentation. Each mathematical argument links to the sources that support it. The exhibit explanations and implementations are original project work informed by those sources; a citation is neither an endorsement nor a record of human review.
Numerical checks, browser checks and source checks are recorded with their scope. Teacher/member mathematical review is a separate status and remains pending until a named review is actually supplied. Reported local timings describe the recorded environment. Performance budgets and accessibility targets are goals for evaluation, rather than evidence that every device or browser has passed.
Read and operate the models in different ways
Read without animation pauses automatic movement while retaining parameter controls and numerical summaries. SVG and HTML views provide a route through the mathematics when WebGL is unavailable. Numeric selectors accompany interactions that would otherwise depend on dragging or selecting a small diagram element. Color is accompanied by labels, values or other visible structure.
Keyboard focus, labeled controls, related error messages, reduced-motion preferences and mobile touch alternatives are part of the design. WCAG 2.2 AA is the accessibility review target; additional motion controls are included deliberately. Individual test records state what was checked. These records do not claim an independent accessibility certification.
The six numerical methods
Topology
Count persistent topological cells, measure angles in the flat faces with a stable vector formula, and sum interior defects and boundary turning with compensated summation.
Domain. The listed valid surface meshes with nondegenerate faces and positive axis scaling. Boundary components are counted explicitly.
Limit. Angle defect is integrated discrete curvature. This exhibit does not estimate a pointwise smooth Gaussian curvature field.
Fourier
Resample the closed contour at equal arc lengths, then calculate a direct complex DFT. The forward transform divides by N; the inverse is unscaled. Both truncation-error quantities are computed separately.
Domain. 64–512 samples using the listed powers of two, with a stated retained term count and DC policy. The even Nyquist frequency is represented as −N/2.
Limit. Complete coefficients reconstruct the samples. Values between samples are trigonometric interpolation, and a sparse reconstruction is not the original continuous drawing.
Geometry
Use the selected metric for lengths and tangent directions for angles. Spherical area has an independent vector expression; hyperbolic area is calculated from angle defect. Disk and hyperboloid distances use equivalent coordinate descriptions.
Domain. Nondegenerate plane triangles, permitted common-hemisphere spherical triangles with short arcs, and points strictly inside the bounded interactive disk domain.
Limit. A Euclidean ruler applied to the disk or hyperboloid picture measures a different quantity. Degenerate triangles have no reported area.
Complex
Retain the unwrapped angle along a nonzero circular path. Evaluate the continued square root or logarithm separately from the selected branch, and check the defining relation at the sampled values.
Domain. The stated positive radii and finite angle range. An analytic branch excludes its cut, even when a display convention assigns a boundary value.
Limit. A three-dimensional surface view is a projection of four real input/output coordinates. The finite visible turn range does not bound the possible logarithm branches.
Spectral
Construct the unnormalized Laplacian from positive-weight edges, diagonalize the real symmetric matrix with bounded cyclic Jacobi sweeps, and evaluate heat through the same numerical eigenbasis at each observation time.
Domain. Undirected, loop-free graphs of at most 64 nodes with nonnegative weights. Zero weight removes a connection; connected bounds require a numerically resolved positive gap.
Limit. Eigenvector signs and bases for repeated eigenvalues are not unique. Disconnected graphs retain separate component means. Observation spacing is not an integration step.
Chaos
Integrate the Lorenz equations with fixed-step fourth-order Runge–Kutta. Keep the initial-perturbation comparison separate from step refinement, and compare refined solutions at common time points.
Domain. The displayed bounded parameter and time ranges, with T/h an integer and no silently shortened final step. Defaults are σ = 10, ρ = 28, β = 8/3, initial state (1,1,1), step 0.005 and observation duration 40.
Limit. The paths are trajectories of model variables. Long-time pointwise agreement is not guaranteed, and the exhibit does not estimate a maximum Lyapunov exponent. RK4 differs from Lorenz’s original numerical procedure.
Mathematical reference library
The sources below support the mathematical content. Specific derivations identify the passages they use.
[M1] Discrete Differential Geometry: An Applied Introduction
Keenan Crane
Version dated 29 January 2025. Chapter 2 and Sections 5.5–5.6 provide the background for triangular surfaces, angle defect and discrete Gauss–Bonnet. The exhibit’s explanations and implementation are original project work. Source checked 16 September 2026.
[M2] Discrete Fourier Transform — NumPy reference
NumPy contributors
Definitions, signed-frequency conventions and normalization options. This project uses a forward factor of 1/N and an unscaled inverse, corresponding to norm="forward" rather than the library default. The browser implementation does not depend on NumPy. Source checked 16 September 2026.
[M3] But what is the Fourier Transform?
Grant Sanderson / 3Blue1Brown
An author-created explanation connecting complex rotation and frequency. Used as a reference for mathematical exposition, not as a source of copied animation or as a substitute for the finite DFT derivation. Source checked 16 September 2026.
[M4] Geometry with an Introduction to Cosmic Topology — Section 5.4
Michael P. Hitchman
Author’s textbook treatment of hyperbolic geometry and triangle angle defect. The parent textbook supplies the surrounding Poincaré disk model and geometric context. Distances and angles in the exhibit use the metric of the named model. Source checked 16 September 2026.
[M5] Complex Analysis: A Visual and Interactive Introduction — Riemann Surfaces
Juan Carlos Ponce Campuzano
Author’s visual treatment of multi-valued complex functions, including separate real-component and imaginary-component surface representations. The project builds its own sampled geometry and explains which coordinates each projection retains. Source checked 16 September 2026.
[M6] Complex Analysis: A Visual and Interactive Introduction — The Logarithmic Function
Juan Carlos Ponce Campuzano
Derivation of logarithm values from exp(w)=z, principal-value conventions and analytic branches on cut domains. The exhibit distinguishes a conventionally assigned boundary value from an analytic branch domain. Source checked 16 September 2026.
[M7] Deterministic Nonperiodic Flow
Edward N. Lorenz
Journal of the Atmospheric Sciences 20 (1963), 130–141; equations and equilibrium analysis on p. 135, default parameter choices and original integration method on p. 136. The project’s fixed-step RK4 solver is its own documented choice and differs from the original double-approximation procedure. Source checked 16 September 2026.
[M8] Spectral and Algebraic Graph Theory
Daniel A. Spielman
Incomplete draft dated 2 April 2025. Sections 1.3, 1.5.4 and 3.1 and Chapter 5 support the Laplacian, positive semidefiniteness, connectivity and example spectra. The heat equation solution and decay estimate are derived in the exhibit from this spectral framework. Source checked 16 September 2026.
[M9] CSE 168 — Materials lecture
UC San Diego CSE 168 course, Spring 2017
PDF page 65 gives the solid-angle tangent ratio, credited to Van Oosterom and Strackee. The project evaluates this ratio with atan2 and retains the denominator sign to select the correct quadrant, including permitted triangles with area above πR². Unit-vector solid angle is multiplied by R² and compared separately with angle excess. Source checked 16 September 2026.
[M10] Hyperbolic Geometry — MA448 lecture notes
Caroline Series
Introduction p. xi (PDF page 12) describes the hyperboloid and Lorentz metric; p. xii relates it to the Klein model. The paired three-dimensional view uses that metric rather than Euclidean surface lengths. Source and locator checked 16 September 2026.
[M11] Geometry III/IV — Lecture notes
Anna Felikson
Durham University, 2020–2021. Section 7.2, Theorem 7.3, p. 113 derives distance from the pseudo-scalar product. On the unit upper hyperboloid this gives cosh d = X₀Y₀ − X₁Y₁ − X₂Y₂. Source checked by direct PDF retrieval on 16 September 2026.
[M12] Runge–Kutta methods
Richard Fitzpatrick
University of Texas at Austin computational physics course, 29 March 2006. Equations (25)–(29) give the four classical RK4 stages and local remainder. The project applies the vector form to its Lorenz system; the model paper is a separate source. Source checked 16 September 2026.