Mathematics gets weird very quickly.

Explore imaginary numbers, ten-dimensional hypercubes, self-intersecting Klein bottles, infinity, impossible geometry, quantum probability, chaos, prime-number mysteries and million-dollar unsolved problems.

ONE OF MATHEMATICS' MOST FAMOUS IDENTITIES

e^iπ + 1 = 0

Euler's identity connects e, i, π, 1 and 0. Imaginary exponentiation turns out to be deeply connected with rotation.

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See a 10D cube

Move from a point to a 10-dimensional hypercube with 1,024 vertices.

Quantum Weirdness

Superposition, qubits, measurement, tunneling, uncertainty and entanglement.

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Klein Bottles

Compare classic bottle and figure-eight immersions.

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Impossible Objects

Pictures that locally make sense but globally cannot exist.

i²

√−1 is useful

Turn the number line into a complex plane.

ℍ

Numbers where AB ≠ BA

Quaternion multiplication depends on order.

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Infinite detail

Zoom into the Mandelbrot set and explore chaos.

∞

Bigger infinities

Some infinite sets genuinely contain more elements than others.

2·3·5

Integer weirdness

Collatz, primes, Ulam spirals and modular patterns.

$1M

Million-dollar problems

Seven legendary Millennium Prize Problems.

Scientific Calculator

Scientific arithmetic with common functions and famous constants.

Calculator History

Click an earlier calculation to restore it.

Graphing Calculator

Plot a function, inspect values and numerically estimate derivatives and nearby roots.

f(x)

X minimum

X maximum

Y minimum

Y maximum

Analyze around x =

Move across graph

Complex Numbers

Numbers can live on a plane rather than a line, and multiplication can become rotation.

Complex Arithmetic

z₁ = a + bi

z₂ = c + di

Argand plane

Powers of i

i⁰ = 1 i¹ = i i² = −1 i³ = −i i⁴ = 1 The cycle repeats every four powers.

Power n

Euler Rotation

e^(iθ) = cos θ + i sin θ

Quaternions

Four-component numbers where multiplication does not generally commute.

q = w + xi + yj + zk

Vector

Quaternion-rotated cube

Multiplication gets strange

i² = j² = k² = ijk = −1 ij = k ji = −k jk = i kj = −i ki = j ik = −j Therefore: i · j ≠ j · i

Dimensional Weirdness

The slider renders the actual selected n-dimensional hypercube. Higher dimensions use a stable bounded linear projection so 7D–10D objects no longer collapse into a tiny star or explode toward infinity.

DIMENSION

Tesseract / 4-cube

4D hypercube projection

0D–3D can be visualized directly or conventionally projected. For 4D and above, this display rotates coordinates in n-dimensional space and maps them to a stable 3D basis before finally projecting to the screen. It is a shadow of the object, not what a higher-dimensional observer would literally see.

Hypercube Anatomy

Every additional dimension doubles the number of vertices.

Impossible Geometry

Perspective lets a flat image imply spatial relationships that cannot all be simultaneously true in ordinary three-dimensional space.

Penrose Triangle

Every corner looks plausible by itself. Taken together, the depth relationships contradict one another.

Exact image requested for the Penrose triangle.

Penrose Stairs

The staircase seems to continually rise or descend while somehow returning to its starting point.

Exact image requested for Penrose stairs.

IKEA Impossible Geometry

Another example of geometry that exploits perspective, occlusion or locally consistent visual information.

Why can your eyes accept this?

A flat image loses depth information. Your brain reconstructs depth using: • perspective • occlusion • shading • edges • familiar shapes Impossible figures deliberately make the local clues believable while making the global interpretation contradictory. The same information-loss issue appears when we project 4D or 10D objects onto a 2D screen.

Topology & Klein Bottles

Topology studies properties preserved by continuous deformation rather than ordinary measurements of distance and angle.

Klein Bottle — Classic Bottle Style

The familiar neck-passes-through-the-wall immersion.

Stable Renderer

Classic Klein bottle immersion

Klein Bottle — Donut / Figure-Eight Style

A different immersion of the same abstract topological surface.

Figure-8

Figure-eight Klein immersion

A Klein bottle cannot be embedded in ordinary 3D Euclidean space without self-intersection. The apparent wall crossing is therefore not a rendering mistake; it is a consequence of representing the surface in three dimensions. The classic bottle renderer itself is now orthographic and auto-fitted, so it does not suffer from the earlier camera-plane explosion.

Möbius Strip

A half-twisted strip has one continuous side and one boundary component.

Möbius strip

Trefoil Knot

One of the simplest nontrivial mathematical knots.

Trefoil knot

Euler Characteristic

χ = V − E + F

Vertices

Edges

Faces

Coffee Mug ≈ Donut

Sphere genus 0 χ = 2 Torus genus 1 χ = 0 Double torus genus 2 χ = −2 For a connected closed orientable surface: χ = 2 − 2g

Quantum Weirdness

Quantum mechanics replaces ordinary deterministic state descriptions with complex probability amplitudes. The interactive displays here illustrate key ideas without pretending to be complete simulations of physical experiments.

These are conceptual educational models. In particular, entanglement correlations do not allow usable faster-than-light communication, and the visualized wavefunctions are simplified one-dimensional examples.

Qubit / Bloch Sphere

A pure qubit can be written as cos(θ/2)|0⟩ + e^(iφ) sin(θ/2)|1⟩.

θ = 90°

φ = 0°

50% P(|0⟩)

50% P(|1⟩)

Bloch sphere projection

Double-Slit Interference

Two coherent path amplitudes combine before probabilities are calculated, producing alternating constructive and destructive interference.

Slit separation

Wavelength

Decoherence

Relative detection probability

Uncertainty Principle

Narrow localization in position requires a broader spread of momentum components.

Position width σₓ

Position vs momentum distributions

Quantum Tunneling

A wavefunction can penetrate a classically forbidden region and emerge with nonzero amplitude.

Barrier width

Barrier height / particle energy ratio

Simplified wave amplitude through barrier

Entanglement Correlation Toy

For an ideal singlet-like pair, a common simplified correlation curve is −cos(Δθ). The outcomes remain individually random even though their statistics are correlated.

Detector A angle

Detector B angle

Ideal correlation curve

Schrödinger's Cat Thought Experiment

The cat was designed as a provocative thought experiment about extending quantum superposition language to macroscopic objects.

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Box closed · state not measured

Amplitude vs Probability

Quantum states use amplitudes. Detection probabilities come from squared magnitudes.

Real amplitude

Imaginary amplitude

Fractals & Chaos

Simple deterministic equations can produce infinite-looking complexity and extreme sensitivity.

Mandelbrot Explorer

Click to zoom.

Maximum iterations

Mandelbrot set

Logistic Map

xₙ₊₁ = r xₙ(1 − xₙ)

r = 3.900

Bifurcation diagram

Number Theory Playground

Integers are simple enough to count with and deep enough to create unsolved mathematical problems.

Prime & Factor Lab

Positive integer

GCD / LCM

Collatz Machine

Even → n/2. Odd → 3n+1. Repeat.

Starting integer

Collatz trajectory

Modular Multiplication

Points

Multiplier

n → kn mod N

Ulam Prime Spiral

Write integers in a square spiral and highlight the primes.

Ulam spiral

Infinity & Sets

Infinite collections behave very differently from finite ones.

Hilbert's Infinite Hotel

Every room is occupied. Another guest arrives. Move Guest n to Room n+1.

0.999... = 1

Number of 9s: 6

Different Infinities

Natural numbers 1, 2, 3, 4, ... Countably infinite. Integers ..., −2, −1, 0, 1, 2, ... Countably infinite. Rational numbers p/q Countably infinite. Real numbers all points on a line Uncountably infinite.

Cantor Set

Remove the middle third repeatedly.

First seven stages

Probability Weirdness

Intuition often performs badly when conditional probability or many possible pairings are involved.

Monty Hall

Simulations

Birthday Paradox

People

Monte Carlo π

Points

Monte Carlo π

Millennium Prize Problems

Seven exceptionally deep mathematical problems. The Poincaré Conjecture is the solved member of the original seven.

P vs NP

Unsolved

Roughly: if a proposed solution can be verified efficiently, can a solution always also be found efficiently?

Riemann Hypothesis

Unsolved

Concerns the location of nontrivial zeros of the Riemann zeta function and their relationship with prime numbers.

Navier–Stokes

Unsolved

Basic existence and smoothness questions for three-dimensional fluid-flow solutions remain unresolved.

Yang–Mills & Mass Gap

Unsolved

Seeks a rigorous mathematical construction of quantum Yang–Mills theory with a positive mass gap.

Hodge Conjecture

Unsolved

Connects topology and algebraic cycles on complex algebraic varieties.

Birch & Swinnerton-Dyer

Unsolved

Connects rational points on elliptic curves to an associated L-function.

Poincaré Conjecture

Solved

Characterizes the 3-sphere among closed simply connected 3-manifolds.

Grigori Perelman proved it using Ricci-flow techniques.

Constants & Oddities

Famous constants, special numbers and numerical curiosities.

3.14159265358979323846...

Euler's number

2.71828182845904523536...

Golden ratio

1.61803398874989484820...

√2

Square root of 2

1.41421356237309504880...

Tau

6.28318530717958647692...

Euler–Mascheroni

0.57721566490153286060...

Kaprekar's 6174

4-digit number

1729

1729 = 1³ + 12³ 1729 = 9³ + 10³ It is the smallest positive integer expressible as two positive cubes in two distinct ways.

Googol / Googolplex

Googol 10¹⁰⁰ Googolplex 10^(10¹⁰⁰) The exponent of a googolplex is itself a googol.

Euler's Identity

e^iπ + 1 = 0