Answer Key

IDU — Aesthetics

Pack A — Answers

#	Question	Answer
1	How many lines of symmetry does a $\text{square}$ have?	4
2	A regular polygon has $6$ sides. How many lines of symmetry does it have?	6
3	State the order of rotational symmetry of a regular $5$-gon.	5
4	Does the capital letter $A$ have rotational symmetry of order > 1?	No (order 1)
5	Which regular polygon tessellates the plane on its own with side length 1?	Equilateral triangle, square, or regular hexagon
6	State the interior angle of a regular hexagon.	$120°$
7	Does a rectangle (not a square) have rotational symmetry? If so, what order?	Yes, order 2
8	Which of these letters have at least one line of symmetry? B, F, M, P, X.	B, M, X
9	The golden ratio $\phi = \dfrac{1 + \sqrt{5}}{2}$. Calculate $\phi$ to 4 d.p.	$\phi \approx 1.6180$
10	The Fibonacci sequence is 1, 1, 2, 3, 5, 8, 13, 21, ... State the next two terms.	34, 55
11	Why does the regular pentagon not tessellate alone?	Interior angle 108° does not divide 360°
12	A regular five-pointed star (pentagram) has how many lines of symmetry and what order of rotational symmetry?	5 lines; rotational order 5
13	State the order of rotational symmetry of the letter N.	2
14	For the Fibonacci sequence 1, 1, 2, 3, 5, 8, 13, ..., compute the ratio of $F_7$ to $F_6$ (3 d.p.).	$13/8 = 1.625$
15	A regular hexagon has side $4$ cm. Find its perimeter.	24 cm
16	A shape is scaled by a linear factor of $3$. By what factor does its area scale?	9
17	A rectangle has sides 5 cm and 8 cm. Is it (approximately) a golden rectangle?	Yes — ratio 8:5 = 1.6 ≈ $\phi$
18	A snowflake has 6-fold rotational symmetry. State (a) the smallest angle of rotation; (b) the number of lines of reflection.	(a) $60°$ (b) 6
19	Find the sum of the interior angles of a regular $12$-gon.	$1800°$
20	A frieze (border pattern) repeats by horizontal translation only — no reflection or rotation. Describe its symmetry group informally.	Translation only (group p1 in frieze notation)
21	A square mosaic tile has 4-fold rotational symmetry and 4 lines of reflection. The total symmetry group has how many elements?	8 (group $D_4$)
22	Show numerically that $\phi - 1 = 1/\phi$ (use $\phi \approx 1.618$).	$\phi - 1 \approx 0.618$; $1/\phi \approx 0.618$
23	The Fibonacci sequence: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55. Compute (a) $F_{10}/F_9$ to 4 d.p. (b) compare to $\phi$.	(a) $55/34 \approx 1.6176$ (b) $\phi \approx 1.6180$ — very close
24	A Voronoi diagram partitions a plane into regions based on proximity to a set of "seed" points. If there are 3 seed points, how many regions does the Voronoi diagram have?	3
25	A regular tessellation uses regular octagons and squares meeting at each vertex. What is the configuration (angles meeting at a vertex)?	4.8.8 (square, octagon, octagon)
26	A "Fibonacci spiral" is approximated by arcs inscribed in squares of sides 1, 1, 2, 3, 5, 8. What is the total arc length (in terms of $\pi$)?	$10\pi$
27	The Koch snowflake starts as an equilateral triangle. Each iteration replaces each side with 4 segments, each $1/3$ the length. After 1 iteration, how many sides does the figure have?	12
28	A snowflake design has 6-fold rotational symmetry but no lines of reflection. How many symmetry elements does it have?	6 (cyclic group $C_6$)
29	A golden rectangle has shorter side $4$ cm. Find its area (to 2 d.p.).	25.89 cm²
30	Classify the symmetry of the yin-yang symbol.	Rotational order 2; no reflective symmetry (chirality)
31	The Parthenon's façade fits roughly inside a golden rectangle of width 30.88 m. Estimate the height (2 d.p.).	19.09 m
32	Show that a regular octagon cannot tessellate the plane on its own.	$3 \times 135 = 405° > 360°$; no way to fit
33	Two seed points are at $(0, 0)$ and $(6, 0)$. Find the equation of the boundary between their Voronoi regions.	$x = 3$
34	Use $\phi \approx 1.618$ to estimate $F_{10}$ via $F_n \approx \dfrac{\phi^n}{\sqrt{5}}$.	About 55.0 (actual $F_{10} = 55$)
35	The golden ratio satisfies $x^2 = x + 1$. Solve this equation, giving exact values.	$x = \dfrac{1 \pm \sqrt{5}}{2}$
36	In a $4 \times 4$ grid of seeds (4 columns × 4 rows), the Voronoi region of a corner seed is a square. What fraction of the total grid does it occupy?	$1/16$ (one cell of 16)
37	An Escher-style tessellation uses a single shape repeated by translation, rotation, and reflection. What's the maximum number of symmetry types it could include?	4 types: translation, rotation, reflection, glide reflection
38	A Penrose tiling uses two shapes ("kite" and "dart") with angles based on $\phi$. Why is this notable?	They tile the plane without periodic repetition (aperiodic)
39	The Koch snowflake has total perimeter that grows without bound as iterations increase, but encloses a finite area. Briefly explain why.	Perimeter $\times 4/3$ per iteration → infinity; area enclosed converges
40	A 6-pointed star (Star of David) has how many lines of symmetry and what rotational order? Hence give its symmetry group name.	6 lines, rotational order 6, group $D_6$ (12 elements)

Pack B — Answers

#	Question	Answer
1	How many lines of symmetry does a $\text{equilateral triangle}$ have?	3
2	A regular polygon has $8$ sides. How many lines of symmetry does it have?	8
3	State the order of rotational symmetry of a regular $7$-gon.	7
4	Does the capital letter $H$ have rotational symmetry of order > 1?	Yes (order 2)
5	Which regular polygon tessellates the plane on its own with side length 1?	Equilateral triangle and square (besides hexagon)
6	State the interior angle of a regular hexagon.	$108°$
7	Does a rectangle (not a square) have rotational symmetry? If so, what order?	Yes, order 2
8	Which of these letters have at least one line of symmetry? B, F, M, P, X.	D, H, T
9	The golden ratio $\phi = \dfrac{1 + \sqrt{5}}{2}$. Calculate $\phi$ to 4 d.p.	$\phi - 1 \approx 0.6180$
10	The Fibonacci sequence is 1, 1, 2, 3, 5, 8, 13, 21, ... State the next two terms.	34, 55, 89
11	Why does the regular pentagon not tessellate alone?	Interior angle $128.57°$ does not divide 360°
12	A regular five-pointed star (pentagram) has how many lines of symmetry and what order of rotational symmetry?	6 lines; rotational order 6
13	State the order of rotational symmetry of the letter N.	2
14	For the Fibonacci sequence 1, 1, 2, 3, 5, 8, 13, ..., compute the ratio of $F_7$ to $F_6$ (3 d.p.).	$21/13 \approx 1.615$
15	A regular hexagon has side $7$ cm. Find its perimeter.	42 cm
16	A shape is scaled by a linear factor of $4$. By what factor does its area scale?	16
17	A rectangle has sides 5 cm and 8 cm. Is it (approximately) a golden rectangle?	Yes — ratio 13:8 = 1.625 ≈ $\phi$
18	A snowflake has 6-fold rotational symmetry. State (a) the smallest angle of rotation; (b) the number of lines of reflection.	(a) $45°$ (b) 8
19	Find the sum of the interior angles of a regular $20$-gon.	$3240°$
20	A frieze (border pattern) repeats by horizontal translation only — no reflection or rotation. Describe its symmetry group informally.	Translation + horizontal reflection
21	A square mosaic tile has 4-fold rotational symmetry and 4 lines of reflection. The total symmetry group has how many elements?	12 (group $D_6$)
22	Show numerically that $\phi - 1 = 1/\phi$ (use $\phi \approx 1.618$).	$\phi^2 \approx 2.618 = \phi + 1$
23	The Fibonacci sequence: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55. Compute (a) $F_{10}/F_9$ to 4 d.p. (b) compare to $\phi$.	(a) $89/55 \approx 1.6182$ (b) very close to $\phi$
24	A Voronoi diagram partitions a plane into regions based on proximity to a set of "seed" points. If there are 3 seed points, how many regions does the Voronoi diagram have?	7
25	A regular tessellation uses regular octagons and squares meeting at each vertex. What is the configuration (angles meeting at a vertex)?	3.6.3.6 (alternating triangles and hexagons)
26	A "Fibonacci spiral" is approximated by arcs inscribed in squares of sides 1, 1, 2, 3, 5, 8. What is the total arc length (in terms of $\pi$)?	$\dfrac{33\pi}{2}$
27	The Koch snowflake starts as an equilateral triangle. Each iteration replaces each side with 4 segments, each $1/3$ the length. After 1 iteration, how many sides does the figure have?	48
28	A snowflake design has 6-fold rotational symmetry but no lines of reflection. How many symmetry elements does it have?	5 ($C_5$)
29	A golden rectangle has shorter side $6$ cm. Find its area (to 2 d.p.).	58.25 cm²
30	Classify the symmetry of the yin-yang symbol.	Rotational order 2; no reflection (chirality)
31	The Parthenon's façade fits roughly inside a golden rectangle of width 30.88 m. Estimate the height (2 d.p.).	5.82 m
32	Show that a regular octagon cannot tessellate the plane on its own.	Heptagon interior $\approx 128.57°$; doesn't divide 360°
33	Two seed points are at $(0, 0)$ and $(10, 0)$. Find the equation of the boundary between their Voronoi regions.	$x = 5$
34	Use $\phi \approx 1.618$ to estimate $F_{10}$ via $F_n \approx \dfrac{\phi^n}{\sqrt{5}}$.	About 144.0 (actual $F_{12} = 144$)
35	The golden ratio satisfies $x^2 = x + 1$. Solve this equation, giving exact values.	$x = \dfrac{1 \pm \sqrt{5}}{2}$
36	In a $4 \times 4$ grid of seeds (4 columns × 4 rows), the Voronoi region of a corner seed is a square. What fraction of the total grid does it occupy?	$1/25$
37	An Escher-style tessellation uses a single shape repeated by translation, rotation, and reflection. What's the maximum number of symmetry types it could include?	Translation, rotation, reflection, glide reflection
38	A Penrose tiling uses two shapes ("kite" and "dart") with angles based on $\phi$. Why is this notable?	No translational symmetry — pattern never exactly repeats
39	The Koch snowflake has total perimeter that grows without bound as iterations increase, but encloses a finite area. Briefly explain why.	Area $\times 3/4$ per iteration → 0
40	A 6-pointed star (Star of David) has how many lines of symmetry and what rotational order? Hence give its symmetry group name.	10 lines, order 10, group $D_{10}$ (20 elements)

Problems — Worked Solutions

**Golden rectangle construction.** A golden rectangle has the property that, when a square is removed from one end, the remaining rectangle is also golden (similar to the original). (a) Set up the proportion that defines this property. (b) Solve to find the side ratio $\phi$. (c) Construct a golden rectangle with shorter side 6 cm. Find the longer side and verify the property.

Answer

(a) $\frac{a+b}{a} = \frac{a}{b}$ (b) $\phi = \frac{1+\sqrt{5}}{2}$ (c) Longer side ≈ 9.71 cm

(a) Let the golden rectangle have shorter side $a$ and longer side $a + b$ (where $b$ is the remainder after removing the square of side $a$). The proportion is: $$\frac{a+b}{a} = \frac{a}{b}$$ (b) Let $\phi = \frac{a}{b}$. Then $\frac{a+b}{a} = 1 + \frac{b}{a} = 1 + \frac{1}{\phi}$. Setting this equal to $\phi$: $$\phi = 1 + \frac{1}{\phi} \Rightarrow \phi^2 = \phi + 1 \Rightarrow \phi^2 - \phi - 1 = 0.$$ $\phi = \frac{1 + \sqrt{5}}{2} \approx 1.618$. (c) Shorter side $a = 6$ cm. Longer side $a + b = a\phi = 6 \times 1.618 \approx 9.71$ cm. So $b \approx 3.71$ cm. Check: remaining rectangle is $6 \times 3.71$ cm; ratio $6/3.71 \approx 1.618 = \phi$ ✓.

**Symmetry audit of a logo.** A logo design has the following claimed symmetries: 4-fold rotation and 2 lines of reflection. (a) Is this combination of symmetries possible (consistent)? (b) Sketch a simple design with these symmetries. (c) Add 2 more reflection lines. What rotational symmetry must the design now have?

Answer

(a) No — inconsistent; 4-fold rotation forces 4 reflection lines (or 0) (b) Adjust to 4 reflections (c) Still 4-fold, but now $D_4$

(a) **No.** A rotational symmetry of order $n$ forces either 0 reflection lines (cyclic group $C_n$) or $n$ reflection lines (dihedral group $D_n$). You can't have 4-fold rotation and exactly 2 reflections. **Why?** If you have any reflection $R$ in axis $\ell$ and a rotation $\rho$ of order 4, then the conjugates $\rho R \rho^{-1}, \rho^2 R \rho^{-2}, \rho^3 R \rho^{-3}$ are also reflections in 3 other axes obtained by rotating $\ell$ by 90°, 180°, 270°. So you must have 4 reflection lines, not 2. (b) A design with $D_4$ symmetry: a square with two diagonal "X" marks. 4 rotations + 4 reflection lines. (c) Adding 2 reflection lines (total 4) to a design with 4-fold rotation gives the full dihedral group $D_4$. The rotational order is still 4 — but the symmetry group is now larger (8 elements: $C_4$ has 4, $D_4$ has 8).

**Fibonacci in nature.** Many plants have leaves arranged in spirals following Fibonacci numbers. (a) The pineapple has scales in two opposing spirals: typically 8 going one way and 13 the other. Calculate the ratio $13/8$ and compare to $\phi$. (b) Sunflowers have 21 and 34 spiral arms (or 34 and 55, depending on size). Calculate both ratios. (c) Why does this happen? Hypothesise (no calculation needed).

Answer

(a) $13/8 = 1.625$, very close to $\phi$ (b) $34/21 ≈ 1.619$; $55/34 ≈ 1.618$ (c) Optimal packing — most efficient seed placement

(a) $13/8 = 1.625$. $\phi \approx 1.618$. Very close. (b) $34/21 = 1.619\overline{0}$. $55/34 = 1.6176\ldots$. Both extremely close to $\phi$. (c) **Hypothesis:** plants arrange leaves/seeds to maximise sunlight or space per unit. Mathematical analysis (Vogel's formula, 1979) shows that the angle that gives the most efficient seed packing without overlap is $\frac{360°}{\phi^2} \approx 137.5°$, the "golden angle." When seeds are placed at this angle, the resulting spirals naturally have counts that are consecutive Fibonacci numbers — because Fibonacci ratios approximate $\phi$ as the values grow. In short: **the golden angle is the most efficient irrational angle for biological packing**, and Fibonacci spirals are a consequence of this.

**Tessellation project.** You want to design a wall pattern using one type of regular polygon. (a) Which regular polygons tessellate alone? List with reasons. (b) What if you combine two different regular polygons (a "semi-regular" tessellation)? Find one example using polygons that meet 3 or 4 around each vertex. (c) Why can't a tessellation use only regular pentagons?

Answer

(a) Triangle (60°), square (90°), hexagon (120°) (b) e.g., 3.6.3.6 (triangle-hexagon) (c) Interior angle 108° doesn't divide 360°

(a) A regular polygon tessellates alone iff its interior angle divides 360° (so the angles meeting at a vertex sum exactly to 360°): - Equilateral triangle: 60° × 6 = 360° ✓ - Square: 90° × 4 = 360° ✓ - Regular hexagon: 120° × 3 = 360° ✓ Other regular polygons have angles that don't divide 360° (e.g. pentagon 108°, heptagon ≈128.57°), so they don't tessellate alone. (b) **Semi-regular tessellations** — there are 8 of these. Examples: - **3.3.4.3.4**: two triangles, square, triangle, square at each vertex. $60+60+90+60+90 = 360°$ ✓. - **3.6.3.6**: triangle-hexagon alternating. $60+120+60+120 = 360°$ ✓. - **4.8.8**: square + two octagons at each vertex. $90+135+135 = 360°$ ✓. (c) Pentagons have interior angle $108°$. $360°/108° = 3.\overline{3}$, not a whole number. Three pentagons leave a $36°$ gap; four would overlap. So they can't tile alone. (Note: *irregular* pentagons can — there are 15 known types of convex pentagon that tessellate.)

**Voronoi diagram in 2D.** Three points are at $A(0, 0)$, $B(6, 0)$, $C(3, 6)$. Find: (a) The equation of the perpendicular bisector of $AB$. (b) The equation of the perpendicular bisector of $AC$. (c) Their intersection — the centre of the Voronoi cell vertex (the "circumcentre" of $\triangle ABC$). (d) Sketch the three Voronoi cells and label them with their seed.

Answer

(a) $x = 3$ (b) $y = -\frac{1}{2}x + \frac{15}{4}$ (c) $(3, 2.25)$

(a) Midpoint of $AB$: $(3, 0)$. $AB$ is along the $x$-axis (gradient 0). Perpendicular bisector is vertical: $x = 3$. (b) Midpoint of $AC$: $(1.5, 3)$. Gradient of $AC$: $\frac{6 - 0}{3 - 0} = 2$. Perpendicular gradient: $-\frac{1}{2}$. Equation through $(1.5, 3)$: $y - 3 = -\frac{1}{2}(x - 1.5)$, i.e. $y = -\frac{1}{2}x + 3.75$. (c) Set $x = 3$ in (b): $y = -1.5 + 3.75 = 2.25$. So intersection at $(3, 2.25)$. This is the **circumcentre** of $\triangle ABC$ — equidistant from all three vertices. It is the meeting point of the Voronoi cell boundaries. (d) **Sketch**: from $(3, 2.25)$, draw rays along each perpendicular bisector. The plane is divided into 3 "wedge-like" cells (unbounded for an outer-edge configuration), one for each seed point.

**Designing a pattern.** Design a tessellating pattern that includes: (a) At least one type of rotational symmetry. (b) At least one line of reflection. Describe your design and identify all its symmetries. Suggested approach: start with a basic regular tile and add motifs.

Answer

Open creative — see exemplar

**Example design.** Take a square grid. Inside each square, draw a smaller square rotated by 45° (a diamond), then a 4-pointed star inscribed in the diamond. **Symmetries identified:** 1. **Rotation**: each square cell has 4-fold rotational symmetry (rotations by 0°, 90°, 180°, 270°). 2. **Reflection**: 4 lines per cell (2 along sides, 2 along diagonals). 3. **Translation**: by the lattice vectors (cell width horizontally and vertically). 4. **Glide reflection**: along the diagonals. This pattern belongs to wallpaper group **p4m** — one of the 17 wallpaper groups. **Student variations:** - Pinwheel pattern: 4-fold rotation, no reflections (group p4). - Brick pattern: translation only (p1). - Honeycomb with motifs: 6-fold rotation (p6 or p6m). **Submission note for Criterion D:** describe the design choices, justify aesthetically, and explain mathematically using the language of symmetry groups.

**Scaling an artwork.** A photograph is in a golden rectangle frame with shorter side 24 cm. You want to enlarge it by 50%. (a) Find the original area. (b) Find the new area. (c) State the scale factor for area in general terms.

Answer

(a) ≈ 932 cm² (b) ≈ 2098 cm² (c) $k^2$ where $k$ is linear scale factor

(a) Shorter side 24 cm, longer side $24 \times \phi \approx 24 \times 1.618 = 38.83$ cm. Area $\approx 24 \times 38.83 \approx 932$ cm². (b) Linear scale factor 1.5 → new sides $36$ and $\approx 58.25$ cm. Area $\approx 36 \times 58.25 \approx 2097$ cm². Or use scale-factor: $932 \times 1.5^2 = 932 \times 2.25 \approx 2097$ cm². (c) For linear scale factor $k$: area scale factor is $k^2$. (For volume in 3D: $k^3$.)

**Symmetry of M.C. Escher's work.** Look at Escher's "Lizard" tessellation (or describe based on description: interlocking lizards in three orientations covering the plane). (a) How many distinct lizard orientations are there? (b) What kind of symmetry transforms one lizard into another? (c) What is the symmetry group's name (if 3-fold rotational symmetry with no reflections)?

Answer

(a) 3 (b) Rotations and translations (c) Wallpaper group p3

(a) Three distinct orientations — typically the lizard appears in three rotational positions, 120° apart. (b) Each orientation is obtained from the others by **rotation by 120°** about specific centres. The pattern as a whole is also invariant under **translation** by the lattice vectors. There are no reflection axes (lizards are chiral — the design has handedness). (c) The wallpaper group is **p3** — 3-fold rotational symmetry, no reflection. One of the 17 wallpaper groups. (Escher famously studied wallpaper symmetries and used most of the 17 groups in his work — a beautiful intersection of mathematics, art, and crystallography.)

**Fibonacci-numbered art.** A digital artist places dots on a grid following the Fibonacci sequence: at coordinates $(1,1), (1,2), (2,3), (3,5), (5,8), (8, 13)$. (a) Plot the points. Describe the pattern (linear, exponential, spiral?). (b) Compute the gradient between consecutive points. What do you notice? (c) Add the next two points and continue.

Answer

(a) Exponentially growing along a "Fibonacci spiral" direction (b) Gradients approach $\phi$ (c) $(13, 21), (21, 34)$

(a) Plotting: the points "race away" from the origin along a curving path. Each point's coordinates are roughly $\phi$ times the previous — this is exponential growth, like the Fibonacci spiral. (b) Gradients between consecutive points: - $(1,1)→(1,2)$: undefined (vertical). - $(1,2)→(2,3)$: $(3-2)/(2-1) = 1$. - $(2,3)→(3,5)$: $(5-3)/(3-2) = 2$. - $(3,5)→(5,8)$: $(8-5)/(5-3) = 1.5$. - $(5,8)→(8,13)$: $(13-8)/(8-5) = 5/3 ≈ 1.667$. Successive gradients $1, 2, 1.5, 1.667, \ldots$ **oscillate around $\phi \approx 1.618$**. (c) Next pair: $(13, 21)$ and $(21, 34)$.

**Fractal feature.** The Sierpinski triangle starts with a solid equilateral triangle. At each step, the middle quarter (smaller triangle pointing down) is removed. (a) After iteration 1, what fraction of the original area is shaded? (b) After iteration 2? (c) Find a formula for the shaded area after $n$ iterations. (d) What happens as $n \to \infty$?

Answer

(a) 3/4 (b) 9/16 (c) $(3/4)^n$ (d) Tends to zero (yet infinitely many pieces remain)

(a) Removing the central triangle (1/4 of the area) leaves $3/4$. (b) Each remaining small triangle (there are 3) has 1/4 removed. So fraction remaining: $\frac{3}{4} \times \frac{3}{4} = \frac{9}{16}$. (c) Iteration $n$: $\left(\frac{3}{4}\right)^n$. (d) As $n \to \infty$: $\left(\frac{3}{4}\right)^n \to 0$. The area tends to **zero**, yet the Sierpinski triangle contains infinitely many disconnected points — a fractal with "fractional dimension" $\log_2 3 \approx 1.585$ (more than a line but less than a plane).

**Cross-curricular project (Visual Arts × Mathematics).** Design a brief outline for a Criterion D investigation: "How do artists use mathematical principles to create aesthetic appeal?" Structure your outline with: (a) Research question. (b) Three specific examples to investigate (e.g., paintings, sculptures, architecture). (c) Mathematical principles you would analyse for each. (d) How you would present your findings.

Answer

See exemplar outline

**(a) Research question.** "How is the golden ratio $\phi$, the Fibonacci sequence, and symmetry used by artists to create aesthetic appeal? Are these mathematical features perceived as 'beautiful' across cultures?" **(b) Three examples.** 1. **Leonardo da Vinci's Mona Lisa**: composition allegedly based on golden rectangles. Investigate the actual proportions; verify or refute the claim. 2. **The Parthenon (architecture)**: golden rectangle in façade dimensions. 3. **Islamic geometric patterns** (e.g., Alhambra tilings): use of symmetry groups (wallpaper groups). **(c) Mathematical principles.** - For Mona Lisa: measure key features (face, body) and compute ratios. Compare to $\phi$. - For Parthenon: measure column-spacing-to-height ratios. - For Islamic patterns: identify symmetry group, count reflection lines, classify by the 17 wallpaper groups. **(d) Presentation.** - Photos with measurements overlaid. - Tables showing ratios computed vs $\phi$. - Diagrams illustrating symmetry types. - Discussion: which artistic choices align with mathematics, which are myth? **Criterion D angle.** Critique whether the maths is descriptive (artists notice $\phi$ post-hoc) or prescriptive (artists deliberately use $\phi$). Compare across cultures.

**Final design challenge.** Create a poster (sketch and describe) that demonstrates: (a) At least 3 mathematical concepts from this unit. (b) Aesthetic considerations (use of colour, balance, rhythm). (c) A short written justification (200 words max) explaining your mathematical choices. Be specific: which symmetry group, which proportion, which scaling factor.

Answer

Open creative — see exemplar response

**Exemplar poster outline.** **Visual elements:** - Central motif: a **golden rectangle** (sides 21 cm × 13 cm, ratio close to $\phi$). - Inside: a **Fibonacci spiral** drawn through quarter-circles in nested squares of sides 1, 1, 2, 3, 5, 8, 13. - Frame around the rectangle: a **tessellation of equilateral triangles and squares** (3.3.4.3.4 semi-regular tiling). - Colour scheme: monochrome blues from light to dark, following a Fibonacci spacing (1, 1, 2, 3, 5 shades of blue intensity). - A **6-fold symmetric snowflake** in the top-right corner, suggesting dihedral symmetry $D_6$. **Justification (200 words):** This poster demonstrates three core unit concepts: (1) the golden ratio, (2) Fibonacci numbers and the related spiral, and (3) symmetry in tessellation. The central golden rectangle uses sides 21 and 13 — consecutive Fibonacci numbers whose ratio (1.615) approximates $\phi$. Drawing the spiral inside reinforces the link between Fibonacci and $\phi$. The framing tessellation uses the semi-regular 3.3.4.3.4 pattern; the interior angle sum (two triangles + two squares = 360°) gives a valid tiling. The snowflake illustrates dihedral symmetry $D_6$ — 6 rotational and 6 reflective symmetries forming a 12-element group. The colour scheme follows Fibonacci spacing because perceptually that ordering feels balanced — a known result from Gestalt psychology. The overall composition is asymmetric (deliberate, to feel dynamic) but each individual element carries strong internal symmetry. The poster therefore visually communicates the contrast between local mathematical order and global creative balance — a common feature of art that uses mathematics. **Notes for Criterion D evaluation:** demonstrate creativity, justify each mathematical choice, and link to aesthetics.