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Problem-solving Pack
MathematicsYear 10 · IDU — Aesthetics
Problem-solving Pack
Name: _________________________________
Date: _________________ Class: ___________
These problems are designed to challenge you. Read each question carefully. Show all your reasoning — a correct answer without working receives no credit.
1Problem 1 of 12
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 .
(c) Construct a golden rectangle with shorter side 6 cm. Find the longer side and verify the property.
(a) Set up the proportion that defines this property.
(b) Solve to find the side ratio .
(c) Construct a golden rectangle with shorter side 6 cm. Find the longer side and verify the property.
Working space
2Problem 2 of 12
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?
(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?
Working space
3Problem 3 of 12
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 and compare to .
(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).
(a) The pineapple has scales in two opposing spirals: typically 8 going one way and 13 the other. Calculate the ratio and compare to .
(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).
Working space
4Problem 4 of 12
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?
(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?
Working space
5Problem 5 of 12
Voronoi diagram in 2D. Three points are at , , . Find:
(a) The equation of the perpendicular bisector of .
(b) The equation of the perpendicular bisector of .
(c) Their intersection — the centre of the Voronoi cell vertex (the "circumcentre" of ).
(d) Sketch the three Voronoi cells and label them with their seed.
(a) The equation of the perpendicular bisector of .
(b) The equation of the perpendicular bisector of .
(c) Their intersection — the centre of the Voronoi cell vertex (the "circumcentre" of ).
(d) Sketch the three Voronoi cells and label them with their seed.
Working space
6Problem 6 of 12
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.
(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.
Working space
7Problem 7 of 12
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.
(a) Find the original area.
(b) Find the new area.
(c) State the scale factor for area in general terms.
Working space
8Problem 8 of 12
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)?
(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)?
Working space
9Problem 9 of 12
Fibonacci-numbered art. A digital artist places dots on a grid following the Fibonacci sequence: at coordinates .
(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.
(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.
Working space
10Problem 10 of 12
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 iterations.
(d) What happens as ?
(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 iterations.
(d) What happens as ?
Working space
11Problem 11 of 12
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.
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.
Working space
12Problem 12 of 12
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.
(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.
Working space