Roll a die, draw a triangle
Tonight's idea
Draw a triangle with corners A, B and C. Start with any dot inside it.
- Roll a die: 1 or 2 chooses A, 3 or 4 chooses B, and 5 or 6 chooses C.
- Put the next dot halfway from your current dot to the chosen corner.
- Repeat, always moving from the newest dot.
This is a random dynamical system: the rule is fixed, but the die chooses which version of the rule comes next.
Ignore the starting dot. After one move, your dot is in one of the three half-size corner triangles. The open upside-down triangle in the middle is missed. After two moves, the first smaller middle gaps are ruled out. After three moves, the next size is ruled out, and so on. Early dots can sit in finer gaps, so a long run approaches the pattern rather than making it perfect.
The rolls choose the route. The halfway rule chooses the holes.
Watch one: three exact moves
Use A \((0,0)\), B \((128,0)\), C \((0,128)\), and start at \(P_0=(32,32)\). A midpoint is found by averaging the two \(x\)-coordinates and the two \(y\)-coordinates.
Roll 1, so choose A: \(P_1=(16,16)\). Roll 3, so choose B: \(P_2=(72,8)\). Roll 5, so choose C: \(P_3=(36,68)\).
Your turn
- Continue the worked example. A roll of 2 chooses A, then a roll of 4 chooses B. Find \(P_4\) and \(P_5\). Hint: average the coordinates each time.
- Join the midpoints of the three sides of a triangle. This makes three half-size corner triangles and an upside-down middle triangle. Explain why one halfway move must land in a corner triangle. Why can no later dot land inside the middle triangle?
- Try six die rolls on your own triangle, marking each halfway dot. Compare with someone else's route, or try six fresh rolls yourself. Must the routes match? Name one large region that both routes must avoid after the first move.
- Replace the triangle by a square. A four-sided spinner chooses a corner, then you move halfway there. The four half-size corner squares exactly cover the large square. Predict the long-run picture: is there a permanent middle hole? Explain.
Stretch: change the fraction
Return to the triangle, but move two thirds of the way towards the chosen corner each time. Each corner copy now has one third of the original side length. Will the pattern be fuller or thinner than the halfway pattern? What happens to the gaps?
Answers
- Roll 2 chooses A, so \(P_4=((36+0)\div2,(68+0)\div2)=(18,34)\). Roll 4 chooses B, so \(P_5=((18+128)\div2,(34+0)\div2)=(73,17)\).
- Choosing a corner puts the new dot halfway along a line to that corner. All such halfway points lie in that corner's half-size triangle, including its edges. The three corner triangles do not include the inside of the upside-down middle triangle. After every move the dot is in a corner triangle, so the open middle stays empty. A dot can lie on its boundary.
- Drawings vary. No, the routes need not match: different rolls usually give different routes. Both routes avoid the inside of the large upside-down middle triangle after the first move.
- There is no permanent middle hole. The four corner half-squares cover the whole square and meet along their edges, so the first-step argument leaves no forbidden region. With many fair spins, dots spread throughout the square.
Stretch. Thinner. Moving two thirds towards a corner leaves the final one third of the line, so each corner copy has side length one third. The three copies are smaller and farther apart than the half-size copies, leaving larger gaps.
Random choices do not guarantee a neat picture after a small number of rolls. The pattern becomes clearer as more dots are added.