In 1958, Danish carpenter and inventor Godtfred Kirk Christiansen patented the modern LEGO brick with its revolutionary cylindrical stud-and-tube coupling system.

Beyond being the world's most beloved building toy, LEGO bricks are a magnificent tool for exploring solid geometry, discrete 3D coordinate grids, and spatial algorithms!

1. The Discrete Integer Grid: $(x, y, z)$

In conventional 3D computer graphics, positioning objects often requires complex floating-point numbers. In our 3D LEGO Studio, coordinates are completely discrete integers:

• $x$ (Horizontal stud index): Steps left and right along the baseplate ($-16 \dots +16$).
• $z$ (Depth stud index): Steps forward and backward along the baseplate ($-16 \dots +16$).
• $y$ (Vertical layer index): Integer tier count ($y = 0$ rests directly on the baseplate, $y = 1$ is layer 1, $y = 2$ is layer 2, etc.).

2. Brick Proportions & Interlocking Overhangs

LEGO bricks follow precise mathematical ratios:

• Stud Pitch: The horizontal center-to-center spacing between studs is $10\text{ units}$.
• Brick Height: One standard brick is $12\text{ units}$ tall—a classic $6:5$ height-to-width aspect ratio!
• Plates: A LEGO plate is exactly $\frac{1}{3}$ the height of a brick ($4\text{ units}$).
• Interlocking Strength: When a brick on layer $y = 1$ spans across two adjacent bricks on layer $y = 0$, it distributes shear forces across both pillars, producing exceptional structural stability!

3. Solid Geometry & The Sum of Squares $\sum k^2$

Consider building a stepped pyramid where each tier $k$ (from $k = 1$ at the apex to $k = N$ at the base) is a square grid of $k \times k$ bricks.

The total number of bricks in the pyramid is given by the famous Sum of Squares formula: $$V(N) = \sum_{k=1}^N k^2 = 1^2 + 2^2 + 3^2 + \dots + N^2 = \frac{N(N + 1)(2N + 1)}{6}$$ For a 4-tier pyramid ($N = 4$): $$V(4) = 1 + 4 + 9 + 16 = \frac{4 \times 5 \times 9}{6} = 30\text{ bricks}$$

Controlling the LEGO Studio with Lua

• lego_init(size): Creates a green baseplate of $size \times size$ studs (default 32).
• brick(x, y, z, width, length, color): Places a brick at grid position $(x, y, z)$ with stud dimensions $(width \times length)$ and given color.
• plate(x, y, z, width, length, color): Places a thin plate ($\frac{1}{3}$ brick height).
• lego_clear(): Clears all placed bricks.
• Color Palette: "red", "blue", "yellow", "green", "white", "black", "orange", "cyan", "gold", "gray", "darkgray", "sand_yellow".

Move the mouse over a dotted box for more information.

• Interactive 3D Orbit Controls: Left-click and drag to rotate the model, right-click to pan, and scroll the mouse wheel to zoom in and inspect studs!
• Preset Camera Views: Use the viewport buttons to snap instantly to Isometric, Top, or Front angles.
• Live Telemetry HUD: Real-time tally of total bricks placed, total studs (volume), and maximum layer height.
• 📥 Export to LDraw (.ldr): Click Export .ldr to download your coded model and open it in official digital CAD programs like BrickLink Studio or LeoCAD!