IDU — Aesthetics

How many lines of symmetry does a $\text{equilateral triangle}$ have?

A regular polygon has $8$ sides. How many lines of symmetry does it have?

State the order of rotational symmetry of a regular $7$-gon.

Does the capital letter $H$ have rotational symmetry of order > 1?

Which regular polygon tessellates the plane on its own with side length 1?

State the interior angle of a regular hexagon.

Does a rectangle (not a square) have rotational symmetry? If so, what order?

Which of these letters have at least one line of symmetry? B, F, M, P, X.

The golden ratio $\phi = \dfrac{1 + \sqrt{5}}{2}$. Calculate $\phi$ to 4 d.p.

The Fibonacci sequence is 1, 1, 2, 3, 5, 8, 13, 21, ... State the next two terms.

Why does the regular pentagon not tessellate alone?

A regular five-pointed star (pentagram) has how many lines of symmetry and what order of rotational symmetry?

State the order of rotational symmetry of the letter N.

For the Fibonacci sequence 1, 1, 2, 3, 5, 8, 13, ..., compute the ratio of $F_7$ to $F_6$ (3 d.p.).

A regular hexagon has side $7$ cm. Find its perimeter.

A shape is scaled by a linear factor of $4$. By what factor does its area scale?

A rectangle has sides 5 cm and 8 cm. Is it (approximately) a golden rectangle?

A snowflake has 6-fold rotational symmetry. State (a) the smallest angle of rotation; (b) the number of lines of reflection.

Find the sum of the interior angles of a regular $20$-gon.

A frieze (border pattern) repeats by horizontal translation only — no reflection or rotation. Describe its symmetry group informally.

A square mosaic tile has 4-fold rotational symmetry and 4 lines of reflection. The total symmetry group has how many elements?

Show numerically that $\phi - 1 = 1/\phi$ (use $\phi \approx 1.618$).

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 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?

A regular tessellation uses regular octagons and squares meeting at each vertex. What is the configuration (angles meeting at a vertex)?

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 $\pi$)?

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?

A snowflake design has 6-fold rotational symmetry but no lines of reflection. How many symmetry elements does it have?

A golden rectangle has shorter side $6$ cm. Find its area (to 2 d.p.).

Classify the symmetry of the yin-yang symbol.

The Parthenon's façade fits roughly inside a golden rectangle of width 30.88 m. Estimate the height (2 d.p.).

Show that a regular octagon cannot tessellate the plane on its own.

Two seed points are at $(0, 0)$ and $(10, 0)$. Find the equation of the boundary between their Voronoi regions.

Use $\phi \approx 1.618$ to estimate $F_{10}$ via $F_n \approx \dfrac{\phi^n}{\sqrt{5}}$.

The golden ratio satisfies $x^2 = x + 1$. Solve this equation, giving exact values.

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?

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?

A Penrose tiling uses two shapes ("kite" and "dart") with angles based on $\phi$. Why is this notable?

The Koch snowflake has total perimeter that grows without bound as iterations increase, but encloses a finite area. Briefly explain why.

A 6-pointed star (Star of David) has how many lines of symmetry and what rotational order? Hence give its symmetry group name.