November 28, 2025
Zero angles, max chaos
A triangle whose interior angles sum to zero
Zero-angle ‘triangle’ drops; commenters rage, joke, and ask if it’s even real
TLDR: A math post shows a hyperbolic “triangle” with zero total angles, infinite perimeter, and finite area, proving geometry gets wild off flat paper. Commenters argued if it’s even a triangle, dragged flat earthers, and joked that spherical geometers are math’s trolls—pure chaos with real lessons.
A math blog just dropped a brain-bender: a “triangle” in hyperbolic space whose angles add up to zero and whose perimeter is infinite but area is finite. Cue the comments going feral. One reader bluntly declared, “That’s not a triangle,” while others scrambled for explanations. Curious minds asked if this infinite-perimeter/finite-area trick shows up elsewhere, name-dropping the Koch Snowflake, while geometry fans flexed with links and “it’s not fractal, it’s curved-space” explainers.
The post by John D. Cook calmly explains: on curved surfaces, triangles don’t behave like they do on flat paper. On spheres, angles add up to more than π (you can even make a triangle with three right angles on Earth); in hyperbolic space, angles add up to less than π—down to zero—yet the area maxes out. Think edges that stretch toward the boundary forever, so measuring the border never ends, but the patch of space you cover stays limited.
Drama hit peak when someone weaponized it for globe-proof: “Dear flat earthers, draw a giant triangle.” Another commenter dubbed spherical geometers “the trolls of the math world,” and suddenly geometry had main-character energy. Whether you came for the math or the memes, this triangle started a Friday-night flame war—and taught everyone a little curved reality.
Key Points
- •On a unit sphere, the area of a spherical triangle equals the angle sum minus π (triangle excess).
- •In hyperbolic geometry with curvature −1, triangle area equals π minus the angle sum (triangle defect).
- •Small triangles in both spherical and hyperbolic geometry have angle sums near π due to local Euclidean behavior.
- •A hyperbolic triangle can achieve interior angle sum 0, yielding the maximum area π.
- •The illustrated hyperbolic triangle is improper with ideal vertices, has infinite perimeter but finite area, and its area is invariant to the semicircle radii.