Shape Rotation & Symmetry

Source

Lecture: 3 — Coordinate Transformation
Simulation: 1 — Shape rotation and symmetry (lecture-demo)
Concept: A symmetry transformation is a rotation that maps a shape back onto itself; different shapes admit different discrete sets of such angles, while the circle admits a continuous set.
Serves: "Describe spatial rotations as symmetry transformations" and "Recognise rotations as a prototype for more general symmetry transformations."
Inquiry: Lecturer-driven sweep of the shared angle; each shape's symmetry angles are discovered and printed as the slider passes them.
Misconception: None specified for this sim.

Pedagogy principles applied

ID	Principle	How it shows up here
A1	One concept	Only rotational symmetry — no reflection, translation, or velocity vectors.
A2	Every label serves the concept	The reference dot exists solely to track orientation; no decorative shapes.
B1	Standard terms	"Rotational symmetry," "symmetry angle," "order of symmetry."
B4	Descriptive headers	Panels read "Circle," "Equilateral triangle," "Square," etc.
C1	Numbers referenced	The live θ readout and each discovered angle list are the deliverables of the sweep.
C2	Numbers carry units	All angles shown as "θ = 72°", never a bare number.
D1	Visualisation is largest	The five shape panels dominate; slider and lists are subordinate.
D2	Readable labels	Names, angle values, and θ readout all ≥ 12 px.
D3	Self-identifying	Each panel carries its shape name; no external legend needed.
D5	No clipping	Shapes, ghost outlines and dots fit at every θ; angle lists wrap.
D7	Playable from load	Slider is live immediately; "Show order" is an additive overlay only.
D8	Labels, not narration	Explanatory framing lives in the side panels, never on the canvas.
E3	Meaningful extras	The gold pulse signals a symmetry match; ghosts show alignment.
J2	Faithful rotation	Each rendered shape's angle equals the slider value θ exactly.

Teach me

Speed

Guided Inquiry

1 · Five rigid shapes, one angle

Let’s get to know the panels. Each one holds a rigid shape, turned about its own centre through the same angle θ. The dashed outline records where that shape sat at θ = 0°, and the white dot marks one point on the shape, so you can always tell which way it’s facing.

2 · What a turn changes

First, something simple. A rigid shape is turned about its centre through some angle. Which of these do you think is changed by the turn?

3 · The triangle at θ = 120°

Now look closely at the triangle. At θ = 120° its outline lies exactly on its dashed θ = 0° outline. So where do you think its white dot is?

Here’s a handy cue: a shape pulses gold whenever θ is one of its symmetry angles — the moment it sits back on its dashed outline — and that angle is added to the list beneath its panel.

4 · Counting the angles

Time to generalise. Including θ = 0°, how many angles in one full turn do you think bring a regular n-gon back onto its own outline?

5 · Where the marked point can go

Now follow just one point. As θ is swept continuously from 0° to 360°, what path do you think the white dot traces out?

6 · From rotations to boosts

Let’s name what you counted earlier. That count is the shape’s order of rotational symmetry — 3, 4, 5 and 6 here. And every angle maps a circle onto itself, so its symmetry is continuous rather than a list.

Here’s where it gets exciting: a Lorentz boost is the spacetime version of this. It changes an event’s t and x while leaving −t² + x² fixed — and that sign flip between t² and x² is why a boosted event moves along a hyperbola, not a circle.

θ is sweeping now, so watch each symmetry angle get recorded under its shape.

Listen

Controls

▾

Rotation angle θ θ = 0°

0°360°

Rotation by θ (counter-clockwise, about the centre)

Symmetry angles of a regular n-gon

Triangle n=3 → {0°,120°,240°} · Square n=4 → {0°,90°,180°,270°} · Pentagon n=5 → every 72° · Hexagon n=6 → every 60°.

Order of rotational symmetry

The order equals the number of distinct symmetry angles in one full turn.

Length is invariant under rotation

e.g. rotate (3, 4) by 90° → (−4, 3); the distance √(3²+4²) = 5 is unchanged.

The Physics

Rotations & symmetry · what this sim shows, the physics behind it, and every control

At a glance

A symmetry is a change that moves every point and still leaves the object looking the same. This sim turns five rigid shapes through one angle that you control, so you can see which turns are symmetries of which shape. It is also the template for the Lorentz boost later in the course.

What you'll learn

What a rigid turn changes about a shape, and what it leaves alone.
What makes an angle a symmetry angle of a shape.
How the number of symmetry angles depends on the number of corners.
How turning the shape differs from turning the axes, and how a boost reuses the idea.

The triangle panel at θ = 35°. The white dot marks one point; its dashed twin shows where that point started.

What's on screen

Five panels: circle, triangle, square, pentagon and hexagon, all turned together about their own centres.
Dashed outline and dashed dot: where the shape and its marked point sat at θ = 0°.
Solid shape and white dot: the shape turned by θ, and the point it carries. Each shape has its own colour. A panel can turn gold; section 04 says when.
Symmetry angles: the list under each panel.

See itTurn all five shapes to 40°. Watch for: each solid shape rotated counter-clockwise from its dashed outline, with the white dot carried round with it.

The controls

Press ⌖ on a row to see where that control sits in the sim.

Control	What it changes	Range
Rotation angle θ	Turns all five shapes together, counter-clockwise, about their own centres. Dragging it pauses the automatic sweep.	0°–360° opens at 0°
⊕ Show order	Prints each shape's order of rotational symmetry beside its name. It stays locked during the Guided Inquiry until the questions are answered.	on / off
Panels and lists	Each panel shows one shape. The list under it collects the symmetry angles that θ has passed, by sweeping or by dragging.	5 panels
▶ Play / ⏸ Pause	Starts or freezes the automatic sweep of θ.	top bar
Speed	Makes the sweep faster or slower. The angles that matter do not move.	0.25×–4× opens at 1×
↻ Reset	Sets θ back to 0°, empties each list back to its 0° entry, and restarts the sweep in the mode you are in.	top bar

Fixed: the sweep runs at 18° per second at 1× speed, so one full turn takes 20 s.

A turn changes coordinates, not the shape

The sim turns each shape counter-clockwise about its centre through θ, with the screen as the fixed frame. A point (x, y) goes to (x′, y′):

Whatever θ is, the distance from the centre comes out the same:

Each point stays on the circle of its own radius, so that distance alone fixes the path it can take.
Side lengths, corner angles and area are all built from distances, so they survive the turn.
What changes is each point's coordinates: where the shape sits, not what it is.

Example, θ = 40°: the point (0, r) goes to (−r sin 40°, r cos 40°) = (−0.64 r, 0.77 r). Both coordinates changed; √(0.64² + 0.77²) = 1.00, so the distance did not.

See itTurn all five shapes to 40°. Watch for: every solid shape is the same size as its dashed outline, though its white dot has moved.

A turn is a symmetry when the shape lands on its own outline

An angle is a symmetry angle of a shape if the turned shape lies exactly on its dashed θ = 0° outline. The triangle's corners are 120° apart, so turning it by 120° does that.

Every corner moved one place round: the white dot sits on a different corner. The object looks unchanged; its coordinates are not.
θ = 0° counts too. Doing nothing is a symmetry, called the identity.
In the sim a panel pulses gold when θ is within ±1° of one of its symmetry angles, and the angle is added to the list under it. The circle is always gold.

See itTurn the shapes to 120°. Watch for: the Triangle panel pulsing gold, its outline on the dashed one, and its white dot on a different corner.

A regular n-gon has n symmetry angles

The n corners are evenly spaced, so the shape repeats every 360°/n:

Shape	n	Step	Symmetry angles
Triangle	3	120°	0°, 120°, 240°
Square	4	90°	0°, 90°, 180°, 270°
Pentagon	5	72°	0°, 72°, 144°, 216°, 288°
Hexagon	6	60°	0°, 60°, … , 300°

The number of symmetry angles in one full turn is the shape's order of rotational symmetry. Doubling the corners halves the step.
The circle has no step: every angle works, so its order is ∞. That is a continuous symmetry, not a long list.

This slider moves the sim too.

See itPrint the order beside each shape. Watch for: Order = 3, 4, 5, 6 and ∞ appearing beside the five names.

See itLet θ sweep from 0°. Watch for: the Hexagon pulsing gold at 60°, then the Triangle and Hexagon together at 120°.

The sim turns the shape; the lecture turns the axes

The sim does an active rotation: the frame stays put and the shape moves. Lecture 3 does a passive one: the point P stays put, Alice's axes are turned by θ from Bob's, and only P's components change.

Same angle, transposed matrix: turning the axes counter-clockwise changes the components exactly as turning the point clockwise would.
Example: a point at 90° that the sim turns by +40° lands at 130°. Alice's axes turned by +40° give components that place the same point at 50° on her axes.
Both leave √(x² + y²) alone, which is why the sim's lesson carries over to the notes.

See itTurn the shapes by +40°. Watch for: each white dot moving counter-clockwise away from its dashed twin, never clockwise.

A boost is a rotation with a minus sign

A rotation keeps x² + y² fixed. That is positive in every direction, so a point's path closes into a circle and it comes home after 360°. A Lorentz boost keeps the spacetime interval fixed, and the minus sign in it opens the circle into a hyperbola.

The interval is s² = −c²t² + x² (Lecture 10); the Guided Inquiry's last card uses it with c = 1. With γ = cosh ζ and v/c = tanh ζ, a boost is the rotation with cos and sin replaced by cosh and sinh, and the rapidity ζ plays the part of θ:

A boost slides an event along its hyperbola and never brings it back: ζ grows without limit as v → c, so there is no full turn.

See itTurn on from 300° through a full turn. Watch for: θ wrapping from 360° to 0°, with every white dot back on its dashed twin.

Where it matters

Crystals. A periodic crystal can only have 2-, 3-, 4- or 6-fold rotation axes. A 5-fold axis, like the pentagon's, never appears in one.
Conservation laws. If the laws of physics look the same after any rotation, angular momentum is conserved (Noether's theorem).
Relativity. The Lorentz transformations are the boosts together with the rotations. All of them keep the spacetime interval fixed.

Go deeper: symmetry angles add

Turning by θ_j and then by θ_k is turning by θ_j + θ_k. For a regular n-gon the result is again one of its n symmetry angles once you subtract 360° if needed: for the hexagon, 240° + 180° = 420° ≡ 60°. These n turns form a group, the cyclic group of order n. The same rule R(α)R(β) = R(α + β) holds for every rotation, which is why the circle's list never ends.

Summary

In one line. A symmetry turn moves every point but keeps all distances, so the shape looks unchanged. A regular n-gon has n symmetry angles, and a boost is the same idea with a minus sign in the invariant.

Key points

A rigid turn keeps distances, so side lengths, corner angles and area survive; only coordinates change.
A symmetry angle puts the shape back on its own outline; θ = 0° counts.
A regular n-gon has n symmetry angles, spaced 360°/n apart. The circle has all of them.
The sim turns the shape; the lecture turns the axes, with the transposed matrix.

Common mix-ups

"At a symmetry angle nothing has happened." The corners swapped places. The shape looks unchanged; the coordinates of each corner are not.
"The lecture's R(θ) is the matrix this sim uses." The lecture turns the axes and the sim turns the shape: R_passive(θ) = R_active(−θ).

Things to try on your own

Drag θ to 90°, then to 180°, and name the shapes that glow gold at each.
Press Reset, let θ sweep a full turn, and count the entries in each list. Compare with n.
Find the angles between 0° and 360° where the Triangle and the Hexagon glow together.