What this is

A 3D lab for three connected ideas: Fibonacci numbers, the golden ratio, and Pythagorean trees. It is built from the “Fibonacci Squares” picture: a cube made of small cubes that spiral inward, on a blue grid.

Everything you see is worked out live from a few formulas. Nothing is a recording, and every slider changes real numbers you can read in the Math card.

The ideas, in plain words

• Fibonacci numbers: each number is the last two added together: 1, 1, 2, 3, 5, 8, 13… Squares with those sides fit together, and quarter circles through them draw the golden spiral.
• Golden ratio (φ ≈ 1.618): divide one Fibonacci number by the one before it and the answer settles on φ. The spiral grows by φ every quarter turn.
• Golden angle (137.5°): the turn that spreads points out most evenly. The Golden sphere uses it.
• Pythagorean tree: put a square on each side of a right triangle. The two small squares together have the same area as the big one (a² + b² = c²). Do it again on each small square and a tree grows.

How to use it

• Drag the 3D view to turn it.
• Octopus arms: spiral arms of cubes reach from the middle of the cube out to its faces and corners. Turn motion makes every turn move, Cube spin turns each cube, and Pulse sends a wave down each arm. All the other shapes move the same way.
• Simple controls shows only the main sliders. Full controls shows all of them.
• Shape picks the model. Filter picks Wire, Glass or Block.
• Every slider runs −99.9 to +99.9. The number after the arrow is the real value it makes.
• Tree: it grows level by level, then stops at the depth you chose. Press Replay to watch it again. Branch angle 0 is the golden angle (31.7°). Spiral twist sets how 3D the tree is, and the lowest setting gives the flat classic tree.

Clusters

The Clusters shape puts 240 made-up dots in six hidden groups, then runs K-Means to find them. Each step it gives every dot to its nearest centre, then moves each centre to the middle of its dots. It repeats until nothing moves, then stops.

Each centre grows an octopus arm that reaches out to it from the middle of the cube, and the dots take their arm’s colour. K-Means start picks where the centres begin: random dots, K-Means++ (spread apart), or the golden angle (spread evenly on a sphere). Set the arms slider to 6 to match the hidden groups, or try other numbers and watch what changes.

The dots are demo data, not yours, and K-Means is the only cluster method in this version.

Where these ideas show up, and who benefits

• Data science: see K-Means work step by step and why the starting points matter.
• Nature: sunflower seeds, pinecones, shells and branching plants follow the same spirals and angles.
• Computer graphics and games: trees and plants made from a few rules instead of modelled by hand.
• Teaching math and coding: sequences, ratios, recursion and Pythagoras you can see and move.
• Design, art and architecture: golden proportions and spiral layouts.
• Algorithms: Fibonacci numbers appear in some data structures (Fibonacci heap) and in classic recursion exercises.
• Charts: some traders draw Fibonacci levels. Evidence that they predict prices is mixed, so treat them as a drawing aid.
• Product and jewellery design: spiral and lattice shapes. Export 3D writes a GLB file with an iridescent texture on every cube. The animated export bakes the motion frame by frame.

What it is not

It is a visual and teaching tool. It is not a physics simulation and not a prediction tool. It shares its wire look and bipolar sliders with your other labs, such as the EM Lab, the Light Wave Lab and the Golden Torus.