LEGO Geometry Tutorial

These slides were produced for my LEGO geometry tutorial at Skaerbaek Fan Weekend in 2026. Feel free to benefit from the contents. Please credit and cite me at https://www.instagram.com/legoarno/

The last slides provide a preview of my upcoming book The Art and Science of LEGO, by Arno Knobbe, to appear in 2027

Transcript

1. LEGO Geometry a tutorial at Skaerbaek Fan Weekend by Arno Knobbe
2. Rectangle trick
3. The rectangle trick switched diagonals 2 1 1:2 rectangle trick
4. Rectangle ratios and degrees every trick has a ratio n:m the angle is 𝛼 = 2 tan−1 𝑛/𝑚 1:2 53.1° 1:3 36.7° 1:5 22.6°
5. You can rotate left or right
6. Stud-based and corner-based rectangles stud-based 1:2 corner-based 2:4 = 1:2
7. Sugar grid
8. Rectangle trick with a 4x8 plate 4:8 = 1:2, so the angle is again 53.1°
9. A 1:2 sugar grid a.k.a. multi-grid six round plates fit under the rectangle, but the grid continues indefinitely no longer a need for hinge plates
10. Five more grid corners fit underneath you can also use round plates, but bi-directional jumper plates are ‘clutch-reduced’
11. Reflected wedges
12. Wedge plate diagonals are irrational Pythagorean theorem: diagonal is a square root often irrational Reflected wedges
13. Reflected wedges and the sugar grid 1:2 1:2 A sugar grid accepts wedge plates of the same ratio. Two angles are supported
14. Wedge plate ratios 1:1 1:2 1:3 1:4 1:6
15. The 1:2 sugar grid and its dual 2 1 1:2 and 1:3 are duals
16. The 1:2 sugar grid and its dual for a n:m grid, the dual is 𝑚 − 𝑛 : (𝑚 + 𝑛) e.g., for 1:2, the dual is (2-1):(2+1) = 1:3 1 3 and, for 1:3, the dual is (3-1):(3+1) = 2:4 = 1:2 1:2 and 1:3 are duals
17. 1:3 reflected wedges on a 1:2 sugar grid 1:3 1:2
18. 1:2 and 1:3 reflected wedges on a single grid
19. Rocco Buttliere’s SPQR diorama 1:2 sugar grid
20. Duals combine to form 45° the A-frame (15706) forms a good jig
21. 1:4 and 1:6 sugar grids the bigger the ratio, the sparser the grid and the shallower the angle 1:4 ⇔ 3:5 1:6 ⇔ 5:7
22. Rocco Buttliere’s SPQR diorama 1 6
23. Pythagorean triangles
24. Multiplying sugar grids start with one sugar grid think of the squares as new units
25. Multiplying sugar grids start with one sugar grid think of the squares as new units count out another sugar grid example: 1:2⨉1:3 = 5:5
26. Multiplying sugar grids start with one sugar grid think of the squares as new units count out another sugar grid example: 1:2⨉1:3 = 5:5
27. Multiplying sugar grids a sugar grid is a Gaussian integer, multiplication according to those multiply two grids: add their angles, and multiply their sizes 5 5 multiplying creates a new sugar grid
28. Squaring sugar grids multiply a grid with itself example: 1:2⨉1:2 = 3:4 5 5 4 the original distance is 5 so the square is 5, integer. Yay! 3
29. Squaring sugar grids multiply a grid with itself example: 1:2⨉1:2 = 3:4 5
30. Squaring sugar grids multiply a grid with itself example: 1:2⨉1:2 = 3:4
31. Squaring sugar grids multiply a grid with itself example: 1:2⨉1:2 = 3:4 squaring a sugar grid produces a Pythagorean triangle
32. Pythagorean triangles 32 + 42 = 52 122 + 52 = 132 5 4 3 13 5 12
33. Each ratio produces its own Pythagorean triangle • m2 – n2 • 2mn • m2 + n2 1:4 ratio: m2 – n2 = 42 – 12 = 15 2mn = 2⨉4⨉1 = 8 m2 + n2 = 42 + 12 = 17 17 8 15
34. Pythagorean Star Destroyers 1:3 1:2 1:4 (15,8,17) (4,3,5) (3,4,5) 1:6 (35,12,37)
35. Death Star #75419 1:4 17 15 8
36. Ratios and Pythagorean triples n m ratio dual 2⍺ ⍺ triple 1 1 1:1 0:1 45 90 1 2 1:2 1:3 26.6 53.1 (3, 4, 5) 1 4 1:4 3:5 14.0 28.1 (15, 8, 17) 1 6 1:6 5:7 9.5 18.9 (35, 12, 37) 1 12 1:12 11:13 4.8 9.5 (143, 24, 145) 2 3 2:3 1:5 33.7 67.4 (5, 12, 13) 3 4 3:4 1:7 36.9 73.7 (7, 24, 25) 2 5 2:5 3:7 21.8 43.6 (21, 20, 29) 4 5 4:5 1:9 38.7 77.3 (9, 40, 41)
37. Upright geometry
38. Upright sugar grid
39. Upright reflected wedges and the Slope 53 6044 Slope 53 is a Pythagorean triangle 5 4 3
40. Slope 53 and the sugar grid sugar grid above sugar grid below
41. Upright reflected wedges match the length of the plates above and below, and match the amount of filler
42. Pythagorean Quadruples
43. Pythagorean quadruples
44. Pythagorean quadruples (1,2,2; 3) 12 + 22 + 22 = 32
45. Two Pythagorean quadruples (1,2,2; 3) • • • • 1 wide 2 deep 2 high 3 diagonal (4,4,2; 6) • • • • 4 wide 4 deep 2 high 6 diagonal
46. The Art and Science of LEGO by Arno Knobbe