Types of Symmetry

Rotations, Reflections, and Dihedral Groups — Simple Visual Examples

Rotational Symmetry

Rotational symmetry of order ( n ) (cyclic group ( _n )) means a figure looks the same after rotation by ( 360^/ n ).

R example: rotating a point

theta <- 2 * pi / 5
pts <- data.frame(x = 1, y = 0)
for (i in 1:4) {
  R <- matrix(c(cos(theta), -sin(theta), sin(theta), cos(theta)), 2, 2)
  newpt <- as.numeric(R %*% c(pts$x[i], pts$y[i]))
  pts <- rbind(pts, data.frame(x = newpt[1], y = newpt[2]))
}

ggplot(pts, aes(x, y)) +
  geom_point(size = 5, color = "steelblue") +
  geom_path(color = "gray40", linewidth = 0.8) +
  geom_point(data = data.frame(x = 0, y = 0), aes(x, y), shape = 3, size = 3, color = "black") +
  coord_fixed(xlim = c(-1.3, 1.3), ylim = c(-1.3, 1.3)) +
  theme_minimal() +
  labs(title = "Order-5 rotational symmetry")

Points generated by successive 72° rotations (order 5)

p5.js: interactive rotational symmetry

Reflectional Symmetry

Mirror symmetry across a line (usually the vertical or horizontal axis).

R example

base <- data.frame(
  x = c(0, 1, 1.5, 1, 0),
  y = c(0, 0.5, 0, -0.5, 0)
)
mirror <- data.frame(x = -base$x, y = base$y)

ggplot() +
  geom_polygon(data = base, aes(x, y), fill = "coral", alpha = 0.25, color = "coral", linewidth = 1) +
  geom_polygon(data = mirror, aes(x, y), fill = "steelblue", alpha = 0.25, color = "steelblue", linewidth = 1) +
  geom_vline(xintercept = 0, linetype = "dashed", color = "gray40") +
  coord_fixed(xlim = c(-2, 2), ylim = c(-1, 1)) +
  theme_minimal()

A shape and its mirror image (outlines + light fill)

p5.js: draw with mirror symmetry

Click and drag to draw. The sketch mirrors your strokes across the vertical axis.

Dihedral Symmetry (( D_n ))

Dihedral groups combine rotations of order ( n ) with reflections. These are the symmetries of regular polygons.

R example: regular pentagon and its symmetries

n <- 5
angles <- seq(0, 2 * pi, length.out = n + 1)
poly <- data.frame(
  x = cos(angles),
  y = sin(angles)
)

ggplot(poly, aes(x, y)) +
  geom_polygon(fill = "gold", alpha = 0.6, color = "black") +
  coord_fixed() +
  theme_minimal() +
  labs(title = "Dihedral group D5")

p5.js: kaleidoscope-style dihedral drawing

Click and drag. The sketch applies both rotations and reflections of order 6.

Design Notes for Art Students

Rotational only (( _n )): swirling, pinwheel-like, no mirror lines.
Reflection only: bilateral balance (faces, logos, architecture).
Dihedral (( D_n )): the most common decorative symmetry (snowflakes, flowers, rose windows, tiles).

These geometric ideas become the constraints we later impose on iterative and random processes.