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Each box is a system of just three equations, and none of them ever repeats. The path stays inside a fixed region but never closes and never crosses itself.
Three is the minimum. In a flat plane a smooth flow cannot be chaotic, a result known as the Poincaré-Bendixson theorem, so chaos needs a third dimension. Otto Rössler found his shape in 1976 and said he got the idea from watching taffy being pulled and folded.
Start two copies 0.00000001 apart and the gap grows steadily in all four. The pitch of each tone follows its x coordinate.
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A 2×2 matrix A makes the plane flow by x′ = Ax, and each flow is one of nine kinds: points sink, escape, spiral, circle or settle on a line.
The kind is set by the eigenvalues of A, the roots of λ² − (tr A)λ + det A = 0. det A < 0 gives a saddle. Above det A = (tr A)²/4 they are complex: spirals, or a centre if tr A = 0. Below it, nodes; on it, star and degenerate node.
The stable spiral here has tr A = −1, det A ≈ 2.65, so λ = −0.5 ± 1.55i.
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A quadric surface is the zero set of a second-degree polynomial in x, y and z. Up to rotation and translation, nine standard forms cover every genuine curved one.
The names come from the slices. A level plane z = h cuts ellipses from the ellipsoid, the elliptic paraboloid, the cone, both hyperboloids and the elliptic cylinder, hyperbolas from the hyperbolic paraboloid and cylinder, and pairs of lines from the parabolic cylinder.
The middle row is one equation, x²/a² + y²/b² − z²/c² = k. At k = 1 every slice is an ellipse, at k = 0 they shrink to the apex, and at k = −1 there are none for |z| < c.
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Shapes can look quite different, depending on your point of view.
v/@TheMathFlow
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If the length of an arc of a circle equals its radius, the angle subtended is defined as one radian.
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“Science cannot tell us a word about why music delights us, of why and how an old song can move us to tears.”
— Erwin Schrödinger
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Bohr's planetary model of the atom yields to quantum probability clouds shaped purely by the numbers n, l, and m.
Spherical s orbitals, dumbbell-shaped p orbitals, and multi-lobed d orbitals emerge as the only allowed geometries. These forms dictate chemical bonds, light absorption, and electron flow in solids. The same mathematics designs semiconductors, lasers, and quantum technologies.
Atomic geometry is the silent architect of modern devices.
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At any point on a curve, there's one circle that fits the curve better than any other. It touches the curve at that point, leans in the same direction, and bends by the same amount.
It tells you how sharply the curve is bending at that exact spot, and which way it's leaning. If the curve is bending sharply, the circle is small. If the curve is nearly straight, the circle is huge.
The osculating circle of the curve.
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"What we observe as material bodies and forces are nothing but shapes and variations in the structure of space and time."
– Erwin Schrödinger
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Pringles chips form a hyperbolic paraboloid: z = x²/a² − y²/b² where x²/a² + y²/b² < 1.
The surface rises parabolically along one axis and falls along the other, producing a saddle that contains two continuous families of straight lines.
Those linear generators give the thin wafer high resistance to bending forces.
Packaging lines use the identical geometry so successive chips interlock in a rigid stack that survives the axial compression inside a sealed cylindrical can from factory to shelf.
