Alien Math · Toys
Three toys that shouldn’t work
A room with mirrored walls that a candle can’t fully light. A weight that rises when you cut the string holding it. A die with fourteen faces that only ever lands on one. Each one runs in your browser, one step at a time.
None of these is a new discovery. The room is our own build from a 1995 recipe, the springs are a 1991 demonstration, and the die is a 2014 shape rebuilt from its published numbers. The new parts are the pictures, an exact in-browser check of the room, and a number for how fragile the die is.
Toy 1 · Light, mirrors, one corner
A room a candle can’t light
In the 1950s Ernst Straus asked whether a room with mirrored walls always lights up completely from a single candle. For polygon rooms the answer is no. Walk through it in five steps.
Every direction, checked
A ray can only land exactly on the spot if its slope is a fraction, so we traced every whole-number direction up to 40 steps across, with exact fractions instead of decimals. In the dark room, 0 of 3,920 reach the spot; every one ends in a corner.
The same recipe, wrong corner
Put the target on the other kind of 45° corner and the check finds it: 258 of 3,920 directions reach it. So the check can say yes.
26 walls, not a record
We grew this room by computer search from Tokarsky’s recipe. It has 26 sides, the same as his first one in 1995. The fewest known is 22.
What the model leaves out: light here is a perfect ray with no width, mirrors are perfect, and a ray that hits a corner exactly stops. A real candle has a flame of finite size, and real light would reach a single point by blurring, so you could never build this. Only an idealised point stays dark.
Toy 2 · Springs, strings, traffic
Cut the string, and the weight goes up
Two springs hang one below the other, joined by a short string, with a weight on the bottom. Two longer strings hang loose beside them, doing nothing. Cut the short string.
Before the cut the springs are in a line, so the whole weight stretches each one. After it, they hang side by side, each holding half. Drag the long strings longer to find where the trick stops working.
The road version, which came first
Four thousand drivers go from start to end. Two roads slow down with traffic (one minute per hundred cars) and two always take 45 minutes. Open a free shortcut in the middle and see what it does to everyone’s trip.
Both are the same sum. The cut string and the closed shortcut each turn a single shared path into two side-by-side ones, and the shared part was the bottleneck.
What the model leaves out: springs are ideal (Hooke’s law, no mass), strings don’t stretch, and everything moves straight up and down. The long strings are drawn beside the springs; in a real build they hang at a slight angle. Drivers all know every trip time and pick the fastest.
Toy 3 · A die made of one material
Fourteen faces, one way to land
A weighted die cheats by hiding lead inside. This one is solid, the same material all the way through, and its shape alone means it can only come to rest on one face. Drop it on any face and it rolls over until it reaches that face (brass). Drag to turn the view.
Resting on its one stable face.
The real shape is a needle: about 4 mm thick for every 100 mm of length. The slider stretches the thin sides so you can see the faces. The rolling order comes from the true shape.
From the paper’s numbers
Reshetov printed the 14 face directions as whole numbers. We cut those planes, got 24 corners, found the centre of mass exactly, and checked each face: only face 11 holds.
All roads lead to one face
Start it on any of the 14 faces and the tipping rule walks it to face 11, in at most 8 tips.
A micron is enough to break it
Print it 10 cm long and move its centre of mass by 1.1 µm the wrong way, and a second face can hold. Reshetov’s paper printed the 15-, 16- and 17-face versions, not this one.
What the model leaves out: the die rolls slowly, one face to the next, with no bouncing or spin. A thrown die carries momentum and can skip past faces, though it can still only stop on the one. It rolls in place, and the floor is perfectly flat and hard.
From the same list
already builtTwo drums, one set of notes
Two differently shaped drums that ring at exactly the same pitches. Strike them in your browser and hear it.
strike the drums →Prior art
Who got there first
The dark spot
- George Tokarsky built the first polygon room with a point a candle can’t reach in 1995, with 26 sides, folding 45° triangles exactly as above. Tokarsky and David Castro each found 24-sided rooms in 1997 (Illumination problem).
- Amit Wolecki’s 2019 room has 22 sides, the fewest known, and nobody knows whether fewer is possible (Quanta, 2024; Wolecki, arXiv 1905.09358). Wikipedia also lists a 20-sided Tokarsky room from 1996; we couldn’t reconcile the two counts from what we read.
- Lelièvre, Monteil and Weiss showed in 2016 that in any polygon whose angles are rational, a candle leaves at most finitely many dark points. Wolecki extended it to pairs of points.
The springs
- Dietrich Braess described the road paradox in 1968.
- Joel Cohen and Paul Horowitz built the springs-and-strings version: Paradoxical behaviour of mechanical and electrical networks, Nature 352, 699–701 (1991).
- The 4,000-driver road numbers are the standard textbook example (Braess’s paradox).
The die
- Conway and Guy described a one-face polyhedron in 1969 with 19 faces; Andras Bezdek found 18 in 2011 (Monostatic polytope).
- Alexander Reshetov, A unistable polyhedron with 14 faces, International Journal of Computational Geometry and Applications (2014). The face directions on this page are from its appendix. He suspected 14 might be the minimum; nobody has shown it.
- The harder cousin, a shape with one resting face and one balancing point, was given explicitly as a polyhedron in September 2026: An explicit mono-monostatic polyhedron (arXiv 2609.07827).
The code behind every number: billiard.py (exact ray tracer), find_room.py and tiles.py (room search), die.py and roll.py (the die), make_data.py (writes rooms.json and die.json).
Next click down
Search for a dark room with fewer than 22 sides using the same folding trick and the same exact check; the search code above is small and runs in seconds. For the die, the useful number is the largest safety margin a printable 14- or 15-face shape can have, because that decides whether a real one ever works.
More toys: three shape puzzles, three sound toys, two things that roll, and the quantum ones (backflow, superoscillation) in the quantum wizard room. Back to the Alien Math wing or the Lab.