Constraint Design and Grid Systems

Deck 2 of 2: Feasible Regions and Ratio-Based Design

Learning Objectives: Deck 2 Focus

Continuing from Deck 1:

3. Apply the modeling cycle to constraint-based design problems
4. Analyze design trade-offs — improving one property affects another
5. Solve design problems involving packaging and typographic grid systems

Also revisiting: translating verbal constraints to geometric inequalities (LO 1)

Every Design Exists Within Limits

Design problems always start with constraints:

• Shipping: cm
• Facility: tank through 3 m door → m
• Shelf: , , cm

Constraint translation: verbal requirement → inequality → feasible region

Translating Verbal Constraints into Geometric Inequalities

Shipping Box: Maximize Volume Under a Sum Constraint

Problem: A shipping company limits packages to cm.
Design a rectangular box with maximum volume.

Strategy: Use the equality case () and ask: how should the 150 cm be divided?

A cube maximizes volume when the dimension sum is fixed.

A Cube Maximizes Volume for Any Fixed Sum

Equal dimensions maximize volume for a fixed dimension sum:

Compare: cm³ — lower.

Water Tank: Both Constraints Are Active

Problem: Cylinder must fit in 3 m × 4 m space. Maximize .

Constraints: m, m

Check-In: Designing a Box for a Shelf

A box must hold ≥ 5,000 cm³ and fit on a shelf 30 × 50 × 25 cm.

Give one valid set of dimensions () and verify each constraint.

Try before advancing.

Answer: One Valid Design (Many Are Correct)

One valid solution: cm, cm, cm

Constraints: Depth 20 ≤ 30 ✓ · Width 25 ≤ 50 ✓ · Height 10 ≤ 25 ✓

Packaging: Counting Cans in a Shipping Box

Can: ∅ 8 cm, h = 12 cm. Box: 48 × 40 × 36 cm.

• 48 ÷ 8 = 6 per row; 40 ÷ 8 = 5 rows; 36 ÷ 12 = 3 layers

Total: cans

Packing Efficiency: What Fraction Is Wasted?

Box volume: cm³

Can volume: cm³

22% wasted — circles never tile a plane perfectly.

Feasible Region and the Optimization Mindset

• Feasible region: all designs satisfying every constraint
• Optimal design: the best point in the feasible region

Two types of optima:

• Boundary — objective is monotone; push dimensions to their limits
• Interior — competing effects balance; use table search or calculus

Geometry Shapes Two-Dimensional Design Too

Geometry doesn't just shape containers — it shapes communication.

Every page, website, and poster uses a geometric grid:

• Column widths calculated from page dimensions
• Margins set in proportional ratios
• Layout proportions based on classical geometric relationships

Column Grid Systems: The Formula

columns, each width , separated by gutters of width :

Solving for column width:

Note: columns → gutters between them, not .

Column Grid: Visual Layout with Four Columns

Four columns, three gutters — gutters always separate columns.

Web Grid: Calculating Column Width

960 px wide, 12 columns, 20 px gutters — find each column width.

11 gutters between 12 columns:

Book Page Margins: Ratio-Based Design

Problem: A 6 in × 9 in book with margin ratios 2:3:4:6 (innerouter:bottom).
Given inner margin = 0.75 in, find all four margins.

Check-In: Computing the Poster Column Width

Poster: 24 in wide, 6 columns, 0.5 in gutters. Find each column width.

1. How many gutters?
2. What is total gutter space?
3. Remaining space for columns?
4. Each column width?

Work through the four steps before advancing.

Answer: Poster Column Width Calculation

6 columns, 0.5 in gutters, 24 in wide

Confirm: ✓

Golden Ratio: Dividing a Width Proportionally

Golden ratio: divides a 30 in poster in ratio (image : text).

Let text ; image :

• Text area: ≈ 11.46 in
• Image area: ≈ 18.54 in

Misconception: Design Has No Unique Right Answer

Common error: Any feasible design is declared "correct" — but feasible designs are not all equal.

Reality:

• Design problems have a feasible region and an objective
• The objective determines which feasible design is best
• Step 1: find a valid answer. Step 2: compare by objective.

Key Takeaways: Decks 1 and 2

✓ Modeling cycle applies to all design problems

✓ Constraints define feasible region; optimization picks the best

✓ Grid formula: W = n·c + (n−1)·g

Watch out:

• Constraint (fixed) vs. objective (minimize)
• Unconstrained optimum may be infeasible
• Find the best design, not just any valid one

This Completes the Modeling with Geometry Cluster

HSG.MG.A.3 pulls together everything from the cluster:

What's next:

• In calculus, these optimization ideas are formalized using derivatives
• In engineering and architecture, this modeling cycle drives real design decisions
• In computer science, constraint satisfaction and optimization algorithms extend these ideas