graphitiEvery shape below has a real, checkable equation — the kind you could plug into any graphing tool and watch the exact same curve appear. None of these are approximations dressed up to look impressive; they're the actual math. Here's a circle, a parabola, a wave, a spiral, and a heart, in order from "you already know this one" to "okay, this one's genuinely delightful."
The circle
the shape you can draw in one breath
This equation is really just the distance formula wearing a disguise. It says: take any point (x, y) on the circle, measure its distance to the center (h, k), and that distance is always exactly r — the radius, no matter which direction you measure in. A circle isn't defined by looking round. It's defined by every single point on it being equally far from one center point. "Equally far in every direction" is the whole shape, compressed into nine symbols.
The parabola
the arc under every thrown ball
If you've ever watched a basketball leave someone's hand, arc up, and drop into the hoop, you've watched a parabola get traced in real time — the path an object takes under steady gravity, once you set aside air resistance. It's also the shape engineers deliberately choose for satellite dishes and flashlight reflectors, because a parabola has a genuinely useful property: every ray that comes in parallel to its axis bounces off and passes through one single point, called the focus. That's not a coincidence dressed up as a fun fact — it's the actual reason dish shapes look the way they do.
The wave
the shape of a single, pure musical note
A sine wave is what a single, perfectly steady musical tone looks like when you plot air pressure over time — smooth, symmetric, repeating forever. Real sound is usually messier than one clean sine wave (most notes are actually several sine waves layered together), but the basic up-down-up-down rhythm underneath almost everything that oscillates — a plucked string, a bouncing spring, a swinging pendulum with a small swing — traces back to this same shape.
The spiral
the shape of a coiled hose, unrolled
This one's written a little differently — in terms of an angle (θ) and a distance from the center (r), instead of x and y — because that's the natural language for anything that spins outward. As θ increases (as you sweep further around), r increases right along with it, at a constant rate. The result is a spiral where each loop is spaced the same distance from the one before it — like a rolled-up tape measure or a coiled garden hose, evenly wound.
There's a cousin worth knowing: the logarithmic spiral, r = a·e^(bθ), where the distance grows multiplicatively instead of steadily — each loop is a consistent percentage bigger than the last, not a fixed amount bigger. Spirals that widen as they go — some snail and mollusk shells, the loose arms of some spiral galaxies — tend to approximate this logarithmic shape rather than the evenly-spaced Archimedean one, though real biological growth is messier than any clean formula and only approximates it.
The heart
the one you're about to screenshot
Here's a genuinely fun fact first: the word "cardioid" — a real named curve, r = a(1 + cos θ) — is Greek for "heart-shaped." But a cardioid actually looks more like a rounded apple or a wide teardrop than the pointy, double-lobed heart you doodle in a notebook margin. The curve that actually nails that shape is a different one entirely, built from layered cosine waves instead:
This is a parametric curve — instead of one equation for y in terms of x, you get two equations that both depend on a third variable, t, which you can think of as "time" or "how far around the loop you are." Plug in every value of t from 0 to a full turn, plot the (x, y) pair each one produces, and the dots trace out an actual, unmistakable heart: two rounded lobes at the top, a notch dipping down between them, tapering to a single point at the bottom. The cosine terms at different frequencies (cos t, cos 2t, cos 3t, cos 4t) are what carve that notch — each one adds a slightly different ripple, and stacked together they pinch the top inward exactly where a heart pinches.
It's not a metaphor and it's not a loose approximation — plot it yourself in any graphing tool and you'll get the same shape every time.
None of these equations were invented to look pretty. They're just what "equally far from a center" or "steady outward push" or "two ripples stacked just right" turns into, once you write it down properly.
Why this matters for your own doodles
The point of all five of these isn't that you need to memorize them. It's that the shapes you draw without thinking — a loop, an arc, a coil, a heart in the corner of a notebook — were never really "just doodles." They were always sitting a short distance from a real equation. That's the entire premise behind what a mathgram is: you draw the shape, and the search finds which of these families (or some combination) it's closest to, then reports back the actual numbers for your specific version.
Want to see what that looks like on a real drawing? Check the gallery, or read how it works end to end.