As a kid, I *loved* making spirographs. I still do. Making them feels more like discovery than creativity, like finding hidden wings in the Mathematical Museum of Art. I have not yet found the point where spirographs no longer surprise me.

The surprising variety of forms generated by spirographs are manifestations of just one equation, the circular path troichoid. The shape of the spirograph depends on the radius of a fixed circle, radius of a cycling circle, and the distance of the pen from the center of the the cycling circle.

\[\begin{align} x (\theta) &= (R - r)\cos\theta + d\cos\left({R - r \over r}\theta\right)\\ y (\theta) &= (R - r)\sin\theta - d\sin\left({R - r \over r}\theta\right) \end{align} \]

Where

Ris the radius of the fixed circle

ris the radius of the cycling circle

dis the distance of the pen from the center of the cycling circle

θis the number of radians the cycling circle travels around the fixed circle

x(θ) is the position ofxafter the cycling circle travelsθradians

y(θ) is the position ofyafter the cycling circle travelsθradians

```
cycling_radius <- 1
fixed_radius <- 3
pen_radius <- 2
d_circle <- tibble(
x0 = c(0, fixed_radius - cycling_radius),
y0 = c(0, 0),
radius = c(fixed_radius, cycling_radius),
r_y = c(0, 0),
r_x = c(-fixed_radius / 2, fixed_radius - 1.5 * cycling_radius),
r_lab = c("Fixed\nRadius", "Cycling\nRadius"),
color = c("black", "royalblue")
)
d_segment <- tibble(
x = c(0, fixed_radius - cycling_radius, fixed_radius - cycling_radius),
y = c(0, 0, 0),
xend = c(-fixed_radius + 0.04, fixed_radius - cycling_radius * 2 + 0.04, fixed_radius - cycling_radius + pen_radius - 0.04),
yend = c(0, 0, 0),
color = c("black", "royalblue", "firebrick")
)
ggplot(data = d_circle) +
theme_void() +
ggforce::geom_circle(
aes(
x0 = x0,
y0 = y0,
r = radius,
color = color),
n = 1000) +
coord_equal() +
geom_text(
aes(
x = r_x,
y = r_y,
label = r_lab,
color = color),
vjust = 0.5,
nudge_y = 0.015,
angle = 0) +
annotate(
x = fixed_radius - cycling_radius + pen_radius / 2,
y = 0.015,
geom = "label",
color = "firebrick",
label = "Pen\nDistance",
label.size = 0,
label.padding = unit(3, "pt")) +
geom_segment(
data = d_segment,
aes(x = x, y = y, xend = xend, yend = yend, color = color),
geom = "segment",
linejoin = "mitre",
arrow = arrow(
length = unit(0.025, "npc"),
type = "closed",
angle = 15)) +
annotate(
x = fixed_radius - cycling_radius,
y = 0,
geom = "point",
color = "royalblue") +
scale_color_identity() +
theme(legend.position = "none")
```

Although I still like making spirographs by hand, I wanted to extend what could be done with the traditional spirograph. I wrote the spiro package in R to make images that would be impossible to create on paper.

I cannot usually predict what will happen when I play with the three primary numbers of the equation. However, once a certain combination strikes me as interesting, I play with cutting it into different color segments to see if something interesting happens. Sometimes I merge many spirographs and spin them to see if the emerging patterns are pleasing.

Here I demonstrate what can be done with spiro package. I would love to see what you can do with it.

```
n <- 25
cc <- ochre_palettes$emu_woman_paired[c(6, 11, 2, 7, 9)] %>%
rep(5)
tibble::tibble(
fixed_radius = n + 2,
cycling_radius = 1:n,
pen_radius = 1:n + 0,
transparency = 0.85,
rotation = pi / 6,
colors = cc,
file = paste0("asdf", 1:n, ".svg")) %>%
purrr::pmap(spiro) %>%
image_merge(
output = "emu_woman_sunset.svg") %>%
image_scale(scale = seq(1, 0.1, length.out = n) ^ 0.85)
```

Color palette selected from Emily Kngwarreye’s *Emu Woman* (1988)

```
k <- 8
crossing(cycling_radius = 1:k, fixed_radius = k * 2 + 1) %>%
rowid_to_column("id") %>%
mutate(
colors = lacroix_palette("Coconut", n = k , "continuous"),
file = paste0("sdfds.", id, ".svg")
) %>%
select(-id) %>%
pmap(
spiro,
points_per_polygon = 2000,
draw_fills = F,
transparency = 0.9) %>%
image_merge(
output = "my_non_canonical_backstory.svg") %>%
add_background()
```

```
rainbow_colors <- hsv(
h = seq(1 / 16, 1, length.out = 16),
s = 0.7,
v = 0.7)
spiro(
fixed_radius = 16,
cycling_radius = 5,
pen_radius = 5,
file = "licorice_donut_vivisection.svg",
color_groups = 16,
color_cycles = 2,
points_per_polygon = 50,
colors = rainbow_colors,
transparency = 0.7) %>%
add_background_gradient(
colors = c("white", "black", "black", "white"),
stops = c(.27, .34, .93, 1),
rounding = 1,
radius = 1)
```

Will We Be As We Were?

```
n <- 10
oslo_colors <- scico(
n = n,
palette = "oslo",
alpha = 0.9) %>%
rev()
spiro(
file = "oslo_aster.svg",
rotation = pi / 6,
points_per_polygon = 100) %>%
image_merge(
output = "oslo_aster.svg",
copies = n) %>%
add_fills(
colors = oslo_colors) %>%
image_scale(
scale = sqrt(0.75 ^ (seq(0, n - 1)))) %>%
image_spin(
rpm = 1:n + 1) %>%
add_background(
color = "black",
rounding = 1) %>%
add_restart()
```

Nothing Shimmers

```
set.seed(105)
k <- 15
bg_colors <- paste0("gray", sample(1:k, k))
bg_stops <- sort(runif(k))
spiro(
fixed_radius = 2 * 13 * 17,
cycling_radius = 3 * 11 * 19,
pen_radius = 171,
file = "but_for_the_darkness_nothing_shimmers.svg",
draw_fills = F,
line_width = 3,
color_groups = 380,
color_cycles = 31,
points_per_polygon = 100,
colors = c(
scico(60 * 2, palette = "lisbon", 0.8),
scico(40 * 2, palette = "cork", 0.25),
scico(20 * 2, palette = "lisbon", 1))) %>%
add_background_gradient(rounding = 0, colors = bg_colors)
```

```
k <- 36
files <- paste0("s", 1:k, ".svg")
pen_radii <- seq(3.8, 1.5, length.out = k)
alphas <- rep_len(c(0.85, rep(0.2, 4)), k)
colors <- rep_len(scico(6, palette = "devon"), k) %>%
alpha(., alpha = alphas)
tibble::tibble(
file = files,
pen_radius = pen_radii,
colors = colors) %>%
purrr::pmap_chr(
spiro,
fixed_radius = 7,
cycling_radius = 4,
rotation = -pi / 10,
points_per_polygon = 500,
draw_fills = T,
xlim = c(-7, 7),
ylim = c(-7, 7)) %>%
image_merge(
output = "youre_my_favorite.svg") %>%
add_lines(colors = c(rep(NA,k - 1), "gray")) %>%
image_rotate(degrees = (1:k / 2.5)) %>%
add_background_gradient(
colors = c(
"#FFFFFF",
"#26588E",
"#E5E3F9",
"#283568",
"#C8C3F3"),
radius = 1,
rounding = 1,
stops = c(0.42,0.93,0.96,0.97,1))
```

```
n <- 20
spiro(
fixed_radius = 4,
cycling_radius = 5,
pen_radius = 1,
file = "suspension_of_disbelief.svg") %>%
image_merge(
copies = n,
output = "suspension_of_disbelief.svg") %>%
add_fills(
transparency = 1 / n,
colors = "blue") %>%
image_scale(scale = seq(1,0.1,length.out = n)) %>%
image_spin(rpm = seq(0.5,10, length.out = n)) %>%
add_restart()
```