Follow curated pathways through mathematical ideas, historical objects, exhibitions and investigations.
Node positions and line lengths organize this navigation. They do not measure mathematical distance, importance or strength of a relationship. The same connections are available in the text list.
From Number to Algebra
Start with position, distinguish zero’s roles, then complete a square.
Lines carry the relationship types listed below; proximity has no mathematical or geographic meaning.
Express a constraint and transform it while preserving its solutions. Compare words, diagrams and modern symbols.
Why these connections exist
Babylonian Mathematics explains Place value. Babylonian Mathematics explores place value through a historical object or procedure.
Zero and Place Value explains Place value. Zero and Place Value explores place value through a historical object or procedure.
Zero and Place Value explains Zero. Zero and Place Value explores zero through a historical object or procedure.
Algebra in Baghdad explains Equations. Algebra in Baghdad explores equations through a historical object or procedure.
Connections between the original exhibits
Select a subject to enter its exhibit, or a numbered connection to read the explanation.
01 · Topology ↔ Geometry Local angles and total curvature
Stretching a valid triangular mesh changes its face angles and the angle defect at individual vertices. For a closed surface, their sum remains 2πχ, with χ determined by the mesh connections. Geometry examines the related statement for a geodesic triangle: its angle sum differs from π by curvature times area when curvature is constant. Together, these results show how local angle measurements can participate in a global constraint. The objects are different: Topology adds defects on flat mesh faces, while Geometry measures triangles in specified smooth metrics. A disk or annulus also requires boundary turning in the mesh identity. Compare the derivations before interpreting a colored curvature field.
Vertex angle defect is integrated discrete curvature, not a pointwise smooth curvature measurement. Boundaries require their own contribution.
02 · Geometry ↔ Complex Preserving angles has conditions
The Poincaré disk draws hyperbolic geometry with the correct angles between intersecting tangents, but Euclidean ruler lengths in the picture are not hyperbolic distances. Disk-preserving Möbius maps provide a concrete connection to complex arithmetic. More generally, a holomorphic function preserves oriented angles locally where its derivative is nonzero: its differential is multiplication by a nonzero complex number, combining rotation with uniform scaling. That condition fails for z² at zero. A logarithm additionally needs a chosen analytic branch domain; assigning a value on a cut does not make it analytic there. Geometry compares metrics, while Complex tracks what a selected branch or a continuously followed value means. Neither comparison makes every complex map angle-preserving everywhere.
Conformality concerns local angles. It does not imply preservation of distances, global injectivity or a single branch on the punctured plane.
03 · Complex ↔ Fourier Rotation becomes an algebraic operation
A unit complex number exp(iθ) represents a rotation through θ. A nonzero coefficient sets its radius and starting phase, so a Fourier term becomes a rotating vector with a precisely specified frequency. Adding the retained vectors reconstructs a sampled contour. Complex uses the same exponential to describe a moving input, but asks a different question: what happens to a square root or logarithm when the input completes a turn? The exponential repeats after 2π, while a continued logarithm remembers the unwrapped angle. Thus Fourier frequencies encode repeated rotations, whereas branch continuation preserves path history that a single point cannot reveal. An epicycle animation illustrates the sum; discrete orthogonality establishes the reconstruction result.
The DFT changes the representation of a finite sample vector. Branch continuation follows values along a path and is not a Fourier reconstruction.
04 · Fourier ↔ Spectral Orthogonal modes on samples and graphs
The DFT expresses a finite sample vector in orthogonal complex exponential modes. A symmetric graph Laplacian supplies an analogous eigenbasis for values attached to graph nodes. The connection is exact on an unweighted cycle with at least three nodes: its sine and cosine eigenvectors come from the discrete Fourier modes, with eigenvalues 2 − 2 cos(2πk/N). On the bridged grids, the connections determine different modes; screen coordinates have no role in that calculation. Fourier studies the error from keeping only some coefficients. Spectral instead uses every computed mode and multiplies its amplitude by exp(−λt) to evaluate heat flow. Both depend on orthogonality, but discarding a mode and letting it decay are different operations.
Integer DFT frequency is not generally a graph eigenvalue. Repeated graph eigenvalues determine an eigenspace whose individual basis vectors are not unique.
[M8] Spectral and Algebraic Graph Theory — §5.5, Theorem 5.5.1: ring-graph sine/cosine eigenvectors and eigenvalues; symmetric spectral framework for the modal heat derivation.
05 · Spectral ↔ Chaos Deterministic evolution can behave differently
Graph heat flow and the Lorenz system both specify deterministic evolution from an initial state. Their structures lead to different comparisons. For a fixed undirected graph with nonnegative weights, heat obeys a linear equation; nonnegative Laplacian eigenvalues produce decaying modes, and each connected component retains its mean. Lorenz includes products of state variables, so it is nonlinear and does not have this fixed modal decay description. Nearby computed Lorenz trajectories can separate under the displayed parameters, even though no random forcing is added. Compare those trajectories at one integration step, then separately compare two step sizes at the same initial state. A finite separation plot does not establish chaos for every parameter choice or guarantee long-term numerical accuracy.
A spectral decay bound has explicit graph assumptions. Observed Lorenz sensitivity is a finite numerical experiment, not a universal instability theorem.
[M7] Deterministic Nonperiodic Flow — pp. 135–136: nonlinear Lorenz equations, equilibria, parameters and numerical discussion. RK4 is this project’s separate method choice.