Factor City

An interactive 3D prime factorization visualizer built with Open3D and NumPy. Each integer in a configurable grid is rendered as a tower of colored blocks — one floor per prime factor (with multiplicity), where each prime has a distinct color. The result is a miniature city whose skyline encodes the arithmetic structure of its buildings.

Prerequisites

This script requires Python 3.10+ and the following packages:

Installation

pip install open3d numpy

Python 3.14 (Windows x64) users: Official Open3D wheels are currently provided up to Python 3.12. A pre-compiled wheel for Python 3.14 on Windows (amd64) is available in the GitHub release assets. Download open3d-0.19.0+22048a72f-cp314-cp314-win_amd64.whl and install it manually before installing the other dependencies:

pip install open3d-0.19.0+22048a72f-cp314-cp314-win_amd64.whl
pip install numpy

For other platforms or Python versions above 3.12, you will need to compile Open3D from source.

Usage

python factor_city.py

No command-line arguments are needed. The application opens a 1400×800 interactive window with a side control panel.

How It Works

Factorization → Architecture

Each number in INITIAL_GRID is decomposed into its prime factors with multiplicity using trial division. For example:

30 = 2 × 3 × 5   →   3-floor tower: Red / Blue / Green
12 = 2 × 2 × 3   →   3-floor tower: Red / Red / Blue
 7 = 7            →   1-floor tower: Yellow

Building height encodes the number of prime factors (Ω(n)), and color encodes which prime occupies each floor, making the multiplicative structure of every number immediately visible at a glance.

Prime Color Palette

Prime	Color	Prime	Color
2	Red	17	Cyan
3	Blue	19	Magenta
5	Green	23	Lime
7	Yellow	29	Sienna
11	Orange	31	Steel blue
13	Purple	>31	Gray

A color-coded legend is rendered to the right of the city grid in the 3D scene.

Animated Delivery Truck

A white sphere orbits the city along a sinusoidal path with a hopping vertical motion, driven by an animation_loop that reschedules itself each frame via post_to_main_thread.

Control Panel

Control	Description
Render 3D Equations	Toggle floating n = p₁ × p₂ × … labels above each rooftop
Run Delivery Truck	Pause or resume the animated truck
Truck Speed	Slider — adjusts animation speed from 0.1× to 3.0×
Building Inspector	Displays factorization details for the last clicked building
Construct Custom Block Tower	Enter any integer ≥ 2 and click Deploy to place it in the next empty grid slot

Building Inspector

Left-clicking any block in the scene triggers a raycast pick. The side panel updates with:

Number : 30
Factors: 2 × 3 × 5
Primes : 2, 3, 5
Floors : 3
Grid   : row 0, col 0

The selected building is highlighted by brightening its block colors.

Customizing the Grid

Edit INITIAL_GRID at the top of the script. Any 2D list of integers works — numbers ≤ 1 are treated as empty lots. The grid can be any rectangular shape.

INITIAL_GRID = [
    [30,  42,  66,  78],
    [12,  18,  20,  28],
    [8,   27,  25,  49],
    [6,   10,  15,  35],
    [2,   3,   5,   7],
]

Block dimensions and spacing are controlled by three constants:

BLOCK_SIZE   = 1.0   # footprint (X and Z)
BLOCK_HEIGHT = 0.5   # height per floor (Y)
GAP          = 0.1   # gap fraction between blocks

Scene Navigation

Action	Control
Rotate	Left-click + drag
Pan	Right-click + drag
Zoom	Scroll wheel
Quit	Q or close window