A Gallery of Mathematical Loading Animations Wasm

The base circle is carved by a sevenfold cosine term, so the trail blooms into a rotating seven-petal ring.

Replacing the sevenfold term with a fivefold term reduces the inner loops, giving the curve a cleaner five-petal rhythm.

A ninefold term packs more inner turns into the same orbit, so the floral ring feels denser and more finely braided.

The radius expands and contracts with cos(7t), so the orbit breathes into repeated petals while staying anchored to a circle.

Using r = a cos(5t) creates five evenly spaced lobes, and the breathing multiplier gently swells each petal in and out.

With k = 2, the cosine radius forms broad opposing petals, and the breathing factor makes the center pulse like the original.

With k = 3, the curve resolves into three rotating petals, and the inner breathing keeps the motion from feeling mathematically rigid.

With k = 4, the petals settle into a balanced cross-like rose, and the breathing core adds the same soft pulse as the original loader.

Different sine frequencies on x and y make the path cross itself repeatedly, producing the woven feel of an oscilloscope trace.

The 1 + sin²t denominator pinches the center while preserving two lobes, so the curve naturally reads as a breathing infinity sign.

The rolling-circle terms create nested turns and offsets, so the path feels like a compact spirograph traced by a machine.

This rolling-circle setup resolves into three large looping petals, all breathing together like a compact spiral flower.

With R = 4, the rolling-circle path settles into four looping petals, rotating and breathing as one ring.

With R = 5, the loop count increases to five petals, giving the spiral flower a denser and more ornate rhythm.

The rolling-circle path splits into six petals, and the whole ring breathes in one unified pulse like the original loader.

Exponential and high-frequency cosine terms stretch the wings unevenly, giving the path its unmistakably fluttering butterfly shape.

Because r = a(1 - cos t) collapses to zero at one side and swells on the other, the curve reads like a soft pulsing heart wave.

Starting from r = a(1 + cos t) and rotating the coordinates turns the textbook cardioid into a more legible upright heart.

The x^(2/3) envelope supplies the heart outline, while sin(bπx) fills its interior with adjustable horizontal ripples.

A fast-growing angle combined with a cosine-modulated radius creates a spiral that opens out and closes cleanly back into itself.

Several sine and cosine components interfere with one another, so the shape keeps mutating like a living waveform.

A circle rolling outside another traces looping petals that reach outward, unlike the inward-coiling hypotrochoid.

The cube of cosine and sine squeeze a circle into a four-pointed star with inward-curving sides.

Three cusps create a rounded triangular shape as the inner circle traces at one-third the outer radius.

Two soft cusps form a kidney shape when the rolling circle has half the fixed circle's radius.

When b exceeds a, the polar radius dips negative, folding the curve inward to create a small inner loop.

Raising the exponent above 2 inflates a circle into a rounded square whose corners soften with the pulse.

The product-of-distances definition creates a peanut-shaped loop that gradually rounds out as the breathing ratio shifts.

Using sine instead of cosine rotates each petal by half a period, and the petal count doubles when k is even.

The Artemis II free-return trajectory: an asymmetric lemniscate with a large Earth-side arc and a tight lunar slingshot loop, plus 16 gravitational perturbation harmonics per point.