The Geometry of Design
Symmetry, transformations, tessellations & frieze patterns — the math hidden in nature, art, and design. Start from zero: see it, name it, work an example, then play with it.
- Symmetries in nature & art
- Geometric transformations
- Tiling & tessellations
- Frieze patterns
- identify & describe reflection and rotational symmetry (line, center & angle of rotation);
- illustrate translation, reflection, rotation, dilation & glide reflection;
- tell which polygons tessellate and generate tessellations (incl. Escher-type);
- identify the transformations that build a frieze pattern.
1 Symmetry in Nature & Art
The big idea. Something is symmetric when you can move it — flip it or turn it — and it looks exactly the same as before. Symmetry is everywhere: a butterfly, a parol (Christmas lantern), a flower, capiz windows, woven banig mats.
Reflection symmetry (line symmetry)
A figure has reflection symmetry if a line cuts it into two halves that are mirror images. That line is the line of symmetry — fold along it and the halves match perfectly.
🪞 Lines-of-Symmetry Explorer — a regular polygon has as many lines as it has sides
Rotational symmetry
A figure has rotational symmetry if you can turn it (less than a full circle) about a fixed point and it lands on itself.
- Center of rotation — the fixed point you turn around.
- Angle of rotation — the smallest turn that maps it onto itself: $\text{angle}=360^\circ\!/n$.
- Order — how many times it matches in one full turn (that's $n$).
🔄 Rotation Explorer — turn the polygon and watch it land on itself
2 Geometric Transformations
A transformation is a rule that moves every point of a figure to a new position, giving an image. There are five you must know:
- Translation — slide in a straight line (no turning or flipping).
- Reflection — flip across a mirror line.
- Rotation — turn about a center by an angle.
- Glide reflection — a reflection followed by a slide along the mirror line (think: footprints).
- Dilation — resize bigger or smaller about a center (shape stays, size changes).
▦ Transformation Explorer — original (dark) → image (red)
Clear example — transform the point $(3,2)$
| Transformation | Rule | $(3,2)$ becomes |
|---|---|---|
| Translate right 4, up 1 | $(x,y)$ → $(x+4,\,y+1)$ | $(7,3)$ |
| Reflect over the y-axis | $(x,y)$ → $(-x,\,y)$ | $(-3,2)$ |
| Rotate 90° (CCW, origin) | $(x,y)$ → $(-y,\,x)$ | $(-2,3)$ |
| Dilate by 2 (origin) | $(x,y)$ → $(2x,\,2y)$ | $(6,4)$ |
| Glide (reflect x-axis, slide right 4) | $(x,y)$ → $(x+4,\,-y)$ | $(7,-2)$ |
3 Tiling & Tessellations
A tessellation (tiling) covers a flat surface with shapes that fit together with no gaps and no overlaps — like floor tiles, a honeycomb, fish scales, or a woven mat.
Which regular polygons tessellate — and why
At every meeting point, the corners (interior angles) must add up to exactly 360°. Only three regular polygons do this on their own:
∡ Angles-Around-a-Point — do the corners fill 360°?
Generating tessellations & Escher-type tilings
Once you have one tile, you build the whole pattern with transformations — mostly translation, sometimes rotation, reflection or glide reflection.
Escher-type tessellations (after artist M.C. Escher) start from a shape that already tiles — say a square — then you cut a piece off one side and attach it to the opposite side (a translation) or an adjacent side (a rotation). Because no area is lost, the new curvy tile still tessellates — and it can look like a bird, fish or lizard.
How to make a simple Escher tile (square) ›
- Start with a square that tessellates.
- Cut a "bump" out of the left edge.
- Translate (slide) that exact bump to the right edge and attach it.
- Do the same from top to bottom. The tile is now irregular but has the same area.
- Repeat the tile by translation — it interlocks with no gaps. Add a face and it becomes art!
4 Frieze Patterns
A frieze pattern is a strip pattern that repeats in one direction — like the border on a malong or t'nalak cloth, a fence design, a tiled wall border, or footprints along a path.
Reading a frieze — which transformations build it?
5 Symmetry of Graphs (Even & Odd Functions)
Now take reflection and rotation onto the coordinate plane. A graph can be symmetric about the y-axis, the origin, or the x-axis — and there's a quick algebra test for each.
🪞 Mirror Explorer — $f(x)=x^2+4$
🔄 Rotation Explorer — $f(x)=x^3-3x$
📐 X-Axis Mirror — the relation $x=y^2$
All three at once: the circle $x^2+y^2=25$
- Replace $x$ with $-x$: $(-x)^2+y^2=25$.
- Since $(-x)^2=x^2$: $x^2+y^2=25$ — identical. ✓
- Replace $y$ with $-y$: $x^2+(-y)^2=25$.
- Since $(-y)^2=y^2$: $x^2+y^2=25$ — identical. ✓
- Replace $x$ → $-x$ and $y$ → $-y$: $(-x)^2+(-y)^2=25$.
- Simplify: $x^2+y^2=25$ — identical. ✓
6 Test Yourself
Lessons and quizzes are kept separate. The full 30-item mastery quiz — covering all four standards plus graph symmetry, with instant feedback and step-by-step solutions — lives on its own page: