Gabriel's Horn
You can fill it, but you can never coat it
a golden trumpet that never ends, which you can fill with paint but never coat
what this teaches: infinity, doubling and halving, area and volume, what a paradox is
What you'll learn
- Why a shape can go on forever and still hold a limited amount
- How halving adds up to a total it never passes, while steady steps never stop growing
- What makes something a paradox, and how looking closer can untangle one
- Why real paint and math paint behave differently
- What a singularity is, and why the dent around a black hole has the same curve as this horn
How to play
- Start with The Horn: drag to spin it and slide Build it longer to keep doubling its length.
- In Paint Race, press Double the horn and watch one pile of paint stop while the other keeps climbing.
- Take The Dive to fly down the throat, past the Moon, until even atoms get stuck.
- Finish with Black Hole: squeeze the Sun down to a point, then press Drop the horn in.
FAQ
What age is this for?
Around 8 and up. It uses doubling and halving rather than calculus, so a curious third grader can follow the idea, and older students can check the numbers in the readout.
Is Gabriel's Horn really a black hole?
No, and the game says so plainly. The horn is a math shape, not a thing in space. The real link is that both come from the same rule, one divided by distance, and both pair something limited with something that runs away without end.
So how can you fill it but not coat it?
Real paint has thickness, so it jams once the tube is one molecule wide. Math paint has no thickness at all, so it can keep spreading thinner forever. The paradox lives in the gap between those two kinds of paint.
Do I need to install anything?
No. It runs in any web browser, it is free, and there is no signup.
For parents and teachers
The doubling table in Paint Race is the heart of this. Each section takes half the paint of the one before to fill, but about the same to coat, which is the whole result with no calculus in sight. For older students, it is y = 1/x spun around the x axis from x = 1, with volume pi and infinite surface area, and it pairs naturally with Integral Lab. The black hole stop is honest about its limits: the funnel draws the pull of gravity rather than the shape of space, and most physicists read the singularity as a sign the equations stop working there, not as something truly infinite.
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