Structured chance and spatial grammar

The first randomness lesson made repeatable editions. This lesson asks a harder question: how can chance make a composition feel surprising and authored? The answer is not “add more random numbers.” It is to make choices at different scales, let small choices inherit from large ones, and decide where chance is not allowed to draw.

1. Lesson: make random choices related
2. Practice: compare chaos with family resemblance
3. Exercise: build a luminous panel grammar

Lesson

Read the structure before the decoration

Course-owner-supplied reference attributed to Zach Lieberman. We use it to ask questions about hierarchy, related variation, panel boundaries, density, and light rather than to infer the original implementation.

The original course diagram turns those observations into a teachable three-level grammar. The empty space is generated deliberately, not left to luck.

A useful generative sketch often has a grammar: a small set of relationships that every seed must respect. Looking at a finished image cannot reveal the artist’s exact code, so we will not pretend to reverse-engineer one. We can still identify transferable visual questions: Where are the large divisions? Which marks seem related? Where does the composition become dense? Where does it stop?

The source for this exercise has three levels:

composition: shared grid, dominant angle, two-color family, one quiet region
region:      warped panel, inherited angle, density band, palette role
mark:        clipped line, small offset, width variation

If every line independently chooses x, y, angle, width, and color, the result usually reads as noise. Here a line inherits most of its identity from its region, and each region inherits from the composition. Chance changes a relationship without erasing it.

Correlation means sharing a cause

Suppose the composition angle is 0.8 radians and the allowed regional change is 0.24. A region may draw a small change such as -0.10:

region angle = composition angle + family shift + small random change
             = 0.80              - 0.12        - 0.10
             = 0.58 radians

Every stroke in that region uses 0.58. The strokes differ in spacing and width, but they still look like one family. Correlation here simply means “these values share a cause.” No statistics course is hiding behind the door.

Use the same pattern for color, scale, speed, or curvature:

const float region_angle = composition_angle + bounded_jitter;
const int mark_color_role = region_color_role;

The important move is deciding which value belongs to which level. Randomness chooses within a range you designed; it does not design the hierarchy for you.

One node lattice prevents accidental cracks

The model first generates one (rows + 1) × (columns + 1) lattice. Adjacent panels point to the same nodes. For a 3 × 3 panel arrangement, that means 16 nodes rather than 36 unrelated corners.

An interior node may move a small amount. Boundary nodes stay on the frame. The model then rejects invalid settings and any panel that folds over itself. This shared ownership matters visually: two almost-equal seam positions produce a bright crack or overlap once wide translucent lines are layered.

Clip in a square, then warp the answer

Each region begins as a local square from (0, 0) to (1, 1). An infinite line through that square is trimmed to the first and last boundary it crosses. Its two local endpoints are then mapped into the panel with bilinear interpolation:

top    = lerp(top_left,    top_right,    local_x)
bottom = lerp(bottom_left, bottom_right, local_x)
point  = lerp(top,         bottom,       local_y)

At local (0, 0) the answer is the panel’s top-left corner. At (1, 1) it is the bottom-right. At (0.5, 0.5) it is the panel center. Because clipping happens in the simple square first, line generation does not need special cases for each warped panel.

Glow can be a stack, not a filter

The starter draws the same segment three times: a wide faint line, a medium faint line, and a narrow bright core. Additive blending lets overlapping light accumulate. The official blend-mode reference explains the renderer state used here.

This is not physically correct blur, and it does not claim to reproduce a particular artwork. It is a portable way to study luminous hierarchy before adding a custom shader. Section 13’s ofFbo technique remains available if you later want an offscreen feedback or post-processing experiment.

Draw dark seams after the light layers. The seam is part of the composition, not an apology for imprecise clipping.

Quiet space is a parameter

One region receives quiet_strokes instead of the active density range. That is a constraint, just like a maximum count or a legal viewport. A generated empty area is more reliable than hoping a random sample happens to leave breathing room.

When reviewing seeds, keep the grammar fixed and ask why one result works better: large-scale balance, direction, density, and voids are more useful answers than “this seed got lucky.” Zach Lieberman’s public dailySketches repository shows the value of sustained iteration; use that as permission to make many small authored studies, not as a source of target compositions to copy.

Practice

Practice the hierarchy before running the section tests.

1. Sort choices by scale

Put each choice in one row of a three-row table: dominant angle, panel corner, region density, individual line width, palette family, one quiet region, and line-spacing jitter. More than one organization can work, but every mark-level choice should have a reason not to belong to its parent.

Then predict the visual difference between these two rules:

A: every stroke draws an angle from 0 to 2*pi
B: composition chooses 0.8; regions vary by at most 0.24; strokes inherit region

2. Build and inspect the solution

Linux or macOS:

scripts/section-16.sh generate --project solution
scripts/section-16.sh build --project solution --configuration Release

Windows Developer PowerShell:

.\scripts\section-16.ps1 generate -Project solution
.\scripts\section-16.ps1 build -Project solution -Configuration Release

Press R and confirm the same seed returns. Press N several times and look for what stays related. Press G to remove the wide translucent layers. Notice that the geometry survives even when the glow disappears.

3. Repair independent angles

In exercises/16-structured-chance/shared/structured_chance_model.cpp, temporarily move the angle draw into the stroke loop and give every stroke a full-circle range. Rebuild and compare. The panels remain but their local family resemblance collapses.

Restore the file before continuing:

git restore -- exercises/16-structured-chance/shared/structured_chance_model.cpp

That command discards every uncommitted change in the named file.

Exercise

Problem: compose a constrained luminous family

Create a seeded panel composition whose visible relationships survive at least 20 seeds. Keep one shared node lattice, bounded panel warping, inherited regional angles, one deliberate quiet region, clipped local lines, density caps, R/N/G controls, and a non-glow view. Change the grid proportions, angle hierarchy, density rhythm, line grammar, seams, and palette so your result differs in structure—not only color—from both examples.

Use the section 16 exercise brief as the authority for editable files and constraints. Start in starter/src/design/structured_chance_design.cpp, then change drawing in starter/src/ofApp.cpp.

Run the tests

Linux or macOS:

CXX=g++ tests/run-section-16-tests.sh
CXX=clang++ tests/run-section-16-tests.sh

Windows Developer PowerShell:

.\tests\run-section-16-tests.ps1

The tests check seed replay, exact shared seams, positive panel area, bounded warping and angle inheritance, finite clipped segments, one quiet region, and work caps. They do not grade palette, glow, or composition.

Quick visual check

• Twenty seeds still look like members of one family rather than unrelated demos.
• At least one region stays quiet and no seed fills every available space.
• Panel seams remain dark and continuous at narrow, square, and wide windows.
• G leaves a readable non-glow structure; color is not the only family cue.
• Nothing flashes, and R, N, G, and H are available from the keyboard.
• Your layout, density rhythm, marks, and palette differ from the examples and any cited influence.

If you get stuck

First print one composition angle and two regional angles. If they are unrelated, fix the inheritance before touching color. For seam cracks, print the two node indices used by neighboring panels; they should refer to the same lattice values. For missing lines, test one local center line through (0.5, 0.5) before adding jitter.