数の奥にひそむパターン。

Unlike a sine that rings forever, a wavelet is localized in both time and frequency — slide and stretch it to pinpoint exactly when a feature happens.

Draw a motif in one tile and pick a symmetry — translations, rotations, mirrors and glides flood the whole plane from your stroke.

Each cell is everywhere-closest to its site; its dual is the Delaunay triangulation — drag the sites.

Drag two arrows and watch them add head-to-tail — a vector is a length and a direction you can move around.

Two arrows in 3D: their components as a box, the parallelogram they span, and the angle between them.

Drop a perpendicular: the shadow of one vector onto another is the closest point on its line.

Every point gets an arrow; drop tracer particles and watch them ride the flow along its streamlines.

Sine and cosine are the shadows of a point going around a circle — drag it and watch the waves unspool.

Substitution re-stretches the x-axis so a tangled integral becomes a clean one — and the warped region keeps exactly the same area.

A head reading, writing, and moving on an infinite tape is all that computation needs.

sin²θ + cos²θ = 1 is just Pythagoras on the unit circle — drag θ and watch the two squares always sum to one.

Drag a triangle and watch its four classic centers — three of them ride the same Euler line.

The shortest round trip is brutal to find but easy to improve — watch nearest-neighbor seed a tour, then 2-opt untangle it.

Watch recursion solve the puzzle in exactly 2ⁿ−1 moves — move n−1 aside, shift the big disk, move n−1 back.

Only triangles, squares, and hexagons tile the plane alone — see the angles meet at exactly 360°.

Add polynomial terms one at a time and watch a straight line bend into a curve that hugs the function.

Zoom into any smooth surface and it looks flat — that flat patch is the tangent plane, the surface's local linear approximation.

Spin and flip a regular polygon — every move that leaves it unchanged is an element of the dihedral group Dₙ.

Every positive fraction appears once on a tree grown from mediants — a flawless enumeration of the rationals.

Splines beat one big polynomial — drag the knots and watch a natural cubic spline glide smoothly through them while the single Lagrange curve thrashes.

A gear rolling inside or outside another traces hypotrochoids and epitrochoids — doodle the math.

Locate a point by radius and two angles (r, θ, φ) instead of (x, y, z) — with latitude and longitude arcs.

Spin a curve around the x-axis to sweep out a 3D solid, then stack disks to measure its volume by the disk method.

A first-order ODE assigns a slope to every point; click anywhere to thread the solution curve through it.

Amplitude, frequency, and phase reshape a wave — slide them and watch y = A·sin(ωx + φ) + k bend.

Filters keep some frequencies and kill others — drag the cutoff of a low-, high-, or band-pass and watch the spectrum bins survive or vanish.

Primes are what survive crossing out every multiple — step through the grid and watch the primes glow as their multiples fall.

Watch Dijkstra and BFS grow a frontier of cheapest-known distances across a grid until it floods the goal.

Entropy is average surprise — a fair coin carries one bit, a biased one less, a sure thing nothing at all.

Drag and resize circles to shade unions, intersections and differences — set algebra becomes regions you can see.

Sample a sine too slowly and a high frequency masquerades as a low one — drag the sample rate past Nyquist and watch the alias appear.

Compression swaps repetition for counts — stripes shrink to almost nothing while noise barely budges.

Spin about x, y, z in sequence — swap the order to see that 3D rotations do not commute, and read off the 3×3 matrix.

The n solutions of zⁿ = 1 are n equally spaced points on the unit circle — a perfect regular polygon.

Three ways to a root, three speeds — watch bisection crawl, false-position lean in, and Newton or secant rocket to where f crosses zero.

SOH-CAH-TOA made tangible — drag a right triangle and watch sin, cos, tan stay fixed as you scale it.

Tile the area under a curve with skinny rectangles and watch the estimate snap onto the exact integral as they multiply.

When one quantity changes, the linked ones change at rates tied by the chain rule — slide a ladder, inflate a balloon, fill a cone and watch the partner rate update live.

Drag the scatter, fight the optimum by hand — the best-fit line is the one that shrinks the squared-error squares to a minimum.

Zeros of the denominator become walls — vertical asymptotes — while the degree race fixes the end behavior.

A drunkard’s path spreads like √n — watch many walkers diffuse and the typical distance track the square root of steps.

A rotation packed as an axis and an angle — so two rotations compose by multiplying, with no gimbal lock.

Build ∀ and ∃ statements over a universe of shapes you can edit, evaluate them live, and see why negation swaps the quantifiers.

Every integer right triangle springs from one formula — and the primitive ones grow on an infinite ternary tree.

See a² + b² = c² as squares whose areas literally add up — drag the legs and watch them rearrange.

Defection wins a single round, yet over many repeated games tit-for-tat quietly out-earns everyone.

A power series hugs its function inside a radius and explodes outside — drag x across the interval of convergence.

A polynomial crosses zero exactly at its roots — drag a root and the whole curve reshapes around it.

One polynomial through every point — drag the data and watch the Lagrange curve wiggle wildly between them as the Runge phenomenon takes over.

(r, θ) draws curves a square grid never could — sweep θ and watch roses, cardioids, and spirals trace out.

Area in polar coordinates is swept from pie-slices, not rectangles — ½∫r²dθ. Watch wedges fan out across a rose or cardioid and add up to the exact area.

Rare random events arrive in time — their gaps are exponential and the count per window is Poisson, both built before your eyes.

A plane is every point with a given normal direction; two planes meet in a line — drag the normals and watch the intersection.

A 2D linear system's fate — node, spiral, saddle or center — is read straight off its matrix's eigenvalues.

Fill slots one by one to count arrangements — then divide out the orderings to see why order stops mattering.

Build a shuffle by dragging arrows, then watch it break into disjoint cycles — the hidden skeleton of every permutation.

The nonlinear pendulum's whole story in one plane — gentle swings, full spins, and the separatrix between.

Each number is the sum of the two above — and hidden inside are the binomials, Fibonacci diagonals, and a Sierpinski fractal.

Slice a surface two ways and read off ∂f/∂x and ∂f/∂y — each partial is the slope of a 1D trace.

Watch one (u, v) sheet fold into a sphere, a torus, a Möbius band, a saddle — a surface is a map.

A parametric curve is a moving point (x(t), y(t)) — watch t trace cycloids and Lissajous figures while the velocity vector points along the path.

Carry an arrow around a loop on a sphere keeping it straight — it returns rotated, and that angle is the enclosed curvature.

A parabola is every point equally far from a focus point and a directrix line — drag both and watch it form.

Better rules fit the curve better — lay rectangles, trapezoids, or parabolas under f and watch Simpson crush midpoint and trapezoid on error for the same n.

The bell curve and the 68-95-99.7 rule — drag a point to read its z-score and the probability in the tail.

Slide down the tangent to a root — drag the start and watch the staircase race in, or spin off to chaos.

Colour each starting point by which root Newton's method falls into — the basins interlock in a fractal lace along their borders.

Edit a 2-player payoff matrix and watch the cells where neither player can do better by switching alone light up.

Throw random darts at a square and the fraction landing inside the circle measures π — accuracy by accumulation.

Estimate an integral by throwing darts — count the points landing under the curve and watch the estimate settle as the error shrinks like 1/√N.

Raising a base to growing powers mod n cycles forever — and that one-way scramble is the engine inside RSA.

Modular arithmetic is clock arithmetic — step the hand around a ring of n and watch multiplication weave times-table cardioids.

The map (az+b)/(cz+d) bends the whole plane, yet every circle and line still maps to a circle or line.

A Möbius band has one side and one edge — walk it and you return mirror-flipped; cut it down the middle and it stays in one piece.

Connect every node with the least total wire — Kruskal and Prim greedily add the cheapest safe edge, one at a time.

Solve a game tree backward — assume the opponent plays their best — and watch alpha-beta pruning skip whole branches.

Somewhere on every smooth arc the instantaneous slope equals the average slope — drag the interval and find it.

Push as much water as the pipes allow from source to sink — the maximum flow always equals the minimum cut.

Drag the basis vectors and watch all of space warp — a 2×2 matrix is what it does to the plane.

Multiplying matrices is doing one transformation then another — compose two 2×2 maps and watch why AB ≠ BA.

Apply a stochastic matrix again and again and any start drifts to the same steady state — watch probability flow and settle.

The map of which c keep z²+c from escaping to infinity — an infinitely intricate boundary you can pan and zoom into forever.

Every row, column, and diagonal sums to the same constant — build odd squares by the Siamese method.

Foxes and rabbits chase each other in endless cycles — closed loops that orbit a balance point.

Growth that brakes itself — exponential at first, then leveling off at the carrying capacity.

Lay two compound statements side by side and compare their truth tables row for row — De Morgan, contrapositive and distribution proven by matching columns.

Two perpendicular oscillations draw surprising closed curves — integer frequency ratios tie them into knots.

Two equations are two lines; their crossing is the solution — slide the coefficients to make it unique, parallel, or coincident.

Drag the constraint lines, sweep the objective, and watch the optimum slide to a corner of the feasible polygon.

Mix two vectors with two weights and see every point they can reach — the span fills a plane, or collapses to a line.

Drag a path through a vector field and walk a bead along it, accumulating F·dr — the work done by the field.

A limit is where you are heading, not where you land — approach a hole or a jump from both sides.

An indeterminate 0/0 resolves by racing the slopes — zoom in until numerator and denominator both vanish and their tangent ratio f′/g′ is the limit.

The best-fit line is the projection of your data onto the column space — drag points and watch the residuals shrink.

Any triangle’s sides and angles are locked together — drag the vertices and verify a/sinA = b/sinB = c/sinC live.

Random in the small, predictable in the large — the running average of coin flips converges to its true probability.

Two trivial rules make chaos, then the ant abruptly builds a periodic highway around step 10,000.

A constrained optimum is where the objective’s contour just kisses the constraint curve — the gradients line up.

A handful of rewrite rules, applied over and over, grow plants, dragons and snowflakes — a turtle draws whatever the string says.

Replace every segment with a bump and repeat forever: a curve of infinite perimeter wrapping a finite area, with a fractional dimension.

Some closed loops can never be untangled into a plain circle — render the trefoil, figure-eight and torus knots and read off their crossing number.

Every point of the Mandelbrot set carries its own Julia set — connected inside, scattered dust outside; drag c on the map and watch them morph.

Any simple closed loop splits the plane into a definite inside and outside — shoot a ray and count crossings: odd means inside, even means outside.

Drag the limits a and b — the definite integral is the area swept under the curve, with everything below the axis counting negative.

Prove the base case and that each statement topples the next, and the whole infinite chain falls — mathematical induction made visible.

An infinite region can still have finite area — push the bound toward infinity and watch ∫1/x² settle while ∫1/x grows without bound.

Curves like x² + y² = r² are not functions, yet they still have tangents — drag a point and watch implicit differentiation hand you the slope.

Drag your observed statistic across the null distribution and watch the shaded tail — the p-value — decide reject or fail-to-reject.

A hyperbola is every point whose distances to two foci differ by a constant — two branches between asymptotes.

Frequent symbols earn short codes — merge the two rarest over and over to grow an optimal prefix tree.

Smoothly morph a mug into a torus and a cube into a sphere — topology counts holes, not corners, so the genus never changes.

A full hotel with infinitely many rooms always has space — shift guests up, into even rooms, or pair rooms with buses to fit ever more.

Add 1 + 1/2 + 1/3 + … and watch the sum creep past every bound — diverging, but only as slowly as ln n.

Codewords spread far apart in the bit-cube can shrug off errors — minimum distance sets how many it survives.

A symmetry group sweeps points into orbits — click one and watch |orbit| × |stabilizer| land exactly on the group size.

See circulation around a loop equal the curl summed over every tiny patch inside it — boundary equals interior.

Color the map so no two neighbors match — the fewest colors you can manage is the chromatic number.

Drag dots, click two to wire an edge — and watch degree, paths, and connected components emerge from nothing.

Straighten a skewed basis into a perpendicular one by subtracting off each projection, step by step.

Step size and momentum decide whether descent crawls, overshoots, or rolls smoothly to the bottom of the valley.

Drop a ball on a surface and watch it roll downhill along −∇f — too big a step and it overshoots.

Translate, rotate, reflect, dilate a shape — see which motions keep distances and which scale them.

Stack ever-shrinking tiles for 1 + r + r² + … and watch them fill a finite length toward 1/(1−r) — or blow up.

The straightest path on a curved surface — drag two points and compare the true geodesic with a naive shortcut.

Row reduction zeroes out below each pivot to solve a system — watch the lines slide as the matrix simplifies.

Color a surface by the sign of its curvature — spheres bulge (K>0), saddles split (K<0), cylinders are secretly flat.

A fair coin walks your bankroll between 0 and N — you will hit a wall, and the odds of which one trace a clean straight line.

The accumulated-area function and the original curve are two sides of one coin — the slope of the area is exactly the height of the graph.

Four knobs reshape a parent curve — shift, stretch, and flip a·f(b(x−h))+k while the original ghosts behind it.

f(g(x)) is a two-machine pipeline — drag x, watch it run through g to u, then through f to y.

Every periodic wave is a recipe of harmonics — add them one slider-tick at a time and watch a square wave assemble itself, Gibbs ringing and all.

A tangent–normal–binormal frame rides along a 3D space curve; torsion is how it twists out of its own plane.

Build a signal from sine components and watch the frequency-domain spikes appear — the transform is the dial that finds the pure tones inside.

Drag a curve through a field and watch flux (flow across) accumulate — then contrast it with circulation (flow along).

x = g(x), drawn as a cobweb — drag the start and watch it staircase onto y = x when |g′| < 1, or spiral out to chaos when it does not.

Fibonacci ratios spiral toward φ — build the squares, plot the convergence, and scatter sunflower seeds by the golden angle.

bˣ explodes, log_b(x) crawls — and they are perfect mirror images across the line y = x.

Play a wager thousands of times and watch the running average settle onto the balance point of the distribution — that is E[X].

φ(n) counts how many numbers below n share no factor with it — a ring lights up the survivors.

Trace every edge exactly once — possible iff 0 or 2 vertices have odd degree, the Bridges of Königsberg in your hands.

Numerical integration walks along the slope in little steps — watch the error shrink as you halve the step.

e^(iθ) is a point walking the unit circle — multiplying by it is pure rotation, and at θ=π it lands on −1.

Count vertices, edges and faces of any polyhedron: V − E + F is always 2 for a sphere, 0 for a torus — a number that ignores shape.

The greatest common divisor is the largest square that tiles a rectangle — peel off squares and watch the Euclidean algorithm find it.

Three parity bits let a Hamming (7,4) code not just notice a flipped bit but pinpoint and repair it.

The pins-and-string definition: an ellipse is every point whose distances to two foci sum to a constant.

Sweep a probe vector and find the directions a matrix only stretches, never turns.

Tile a surface with Riemann prisms and watch their volume converge to the exact integral as the grid refines.

Drag two vectors and watch one cast a shadow on the other — the dot product is that shadow times the length.

Paint a complex function by its output — hue is the argument, brightness the magnitude; zeros and poles become color wheels.

Watch net flux out of a closed surface equal the divergence summed over the volume inside — sources push it up.

Drag a tiny test box through a field — it swells where divergence is positive and spins with the curl.

Flip between binomial, Poisson, geometric, uniform, exponential and normal — one family of shapes tuned by a couple of dials.

Spin a direction arrow on a contour map — the slope you feel is ∇f·û, peaking when you face uphill.

Sum more dice and the flat distribution of one die rises into a bell — the central limit theorem in your hand.

Three vectors build a slanted box; its volume is the determinant — squeeze them coplanar and it collapses to zero.

Drag dots on a number line and watch the mean tip like a balance while the median holds its ground.

Shrink a secant line until it collapses onto the tangent — the derivative is a limiting slope.

Slide the tangent along a curve and watch its slope trace out a whole new curve — the derivative.

Step around a ring of n points by a generator g — it visits every point exactly when gcd(g, n) = 1.

Drag a point along a curve and watch the circle that best kisses it — its radius is exactly 1/κ, the curvature.

Shrink a tiny loop and watch its circulation-per-area converge to the curl — the spin rate of a paddlewheel at a point.

Slice a solid with a tilting plane: a cone gives the conics, a cube gives triangles up to hexagons.

Two 3D vectors and the arrow perpendicular to both — its length is the area of the parallelogram they span.

Tighten or loosen the cloud and watch r sweep from −1 to +1 — correlation is the cloud’s shape, not the line’s steepness.

Slide a flipped kernel across a signal — multiply, sum, repeat — and watch the output build point by point: the math of blurring and echo.

A convex bowl has one valley everything rolls into; a bumpy one traps descent in scattered local minima.

The rubber band snapped around a point cloud — watch the gift-wrapping algorithm find it step by step.

The ratio and root tests read a series from how fast its terms shrink — pick one and watch the verdict emerge.

Read a surface as a topographic map — level curves connect equal heights, and crowding means steepness.

Peel any number into a continued fraction and its truncations race toward it as the best possible rationals.

In a conservative field, work between two points is path-independent — every route gives the same total, and curl is zero.

Tilt a plane through a double cone and watch the slice morph: circle → ellipse → parabola → hyperbola.

Draw sample after sample and watch ~95% of the intervals straddle the true mean while a stubborn ~5% miss in red.

Conditioning shrinks the world to the given event, then re-measures — watch P(A|B) become A’s share of B, with Monty Hall as a preset.

Drag two complex numbers — adding shifts head-to-tail, multiplying adds angles and multiplies lengths.

A complex-differentiable map is locally a rotation + scale — little squares stay squares, so angles are preserved.

Rearrange the area of ax²+bx+c into a literal square — vertex form and the quadratic formula fall right out.

The 3n+1 problem — halve the evens, triple-plus-one the odds, and every start seems to crash to 1.

The inscribed angle is always half the central angle — drag points on a circle and watch it hold.

A handful of remainders against coprime moduli pin a number down to exactly one spot inside their product.

Jump halfway toward a random corner, over and over — and the Sierpinski triangle appears out of pure randomness.

The same point has different coordinates in different bases — drag a new basis and read the same vector both ways.

Two function machines in series — a nudge in x ripples through g then f, and the rates multiply.

Averages of any distribution become normal — sample a wild shape, take the mean, and watch a clean bell emerge.

Tiny local rules grow wild global complexity — elementary 1D triangles and Conway’s Game of Life.

A group laid bare as its multiplication table — pick two elements and watch a·b light up, with identity, inverses and subgroups shining through.

One sequence counts a dozen things — lattice paths below the diagonal, balanced parentheses, non-crossing chords.

List the reals row by row, then build a number that differs from every one along the diagonal — proof that the reals are a bigger infinity.

Drop needles on lined paper and the fraction that cross a line estimates π — geometry turning randomness into 3.14159…

Toggle inputs through AND, OR, NOT and XOR gates and watch the truth table light up — a circuit is a table and back.

Expand (a+b)ⁿ term by term and watch each coefficient appear as a Pascal-row count of the ways to pick b.

Every yes/no question that halves the field is worth one bit — so finding 1-of-N takes only log₂N of them.

Drag arrows to pair two sets one-to-one — the app flags injective, surjective and bijective, and shows classic infinite pairings like ℕ ↔ even.

Cross a threshold and fixed points are born, collide, or swap stability — the geometry of sudden change.

A Bézier curve is just repeated linear interpolation — watch the de Casteljau lerps collapse to one point.

In data that spans many scales, leading digit 1 shows up ~30% of the time — watch real-feeling numbers obey the logarithmic law.

Updating beliefs with evidence, shown as areas — see why a positive test for a rare disease often means very little.

Sum 1 + 1/4 + 1/9 + … and land on π²/6 — then slide the exponent s to feel ζ(s) and the p-series threshold.

Shade the region trapped between two curves and read its area as the integral of the gap, top minus bottom, across the bounds.

Approximate a curve with a chain of straight segments and watch the total of their tiny hypotenuses converge to the true arc length.

Mutually tangent circles spawn ever-smaller ones forever, by Descartes’ Circle Theorem.

Alternating shrinking terms bounce in and trap the limit — the error never exceeds the very next term.

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