Schotter plots in R | R-bloggers
Translating elements between languages reveals how each language approaches different design tradeoffs, and I think it’s a useful exercise. Having something to translate is the first step. I found a plot I wanted to generate, and some code that reproduced it, so here we go!
I don’t remember how I originally found this page (I didn’t keep a note on it, it seems) and it sat on my to-be-posted pile for too long, so here is the post I intended to write.
This article details the ALGOL code that generates Georg Nees’ 1968 computer-generated art “Schotter”, which shows a grid of squares that are increasingly moved in position and rotation.
1 'BEGIN''COMMENT'SCHOTTER., 2 'REAL'R,PIHALB,PI4T., 3 'INTEGER'I., 4 'PROCEDURE'QUAD., 5 'BEGIN' 6 'REAL'P1,Q1,PSI.,'INTEGER'S., 7 JE1.=5*1/264.,JA1.=-JE1., 8 JE2.=PI4T*(1+I/264).,JA2.=PI4T*(1-I/264)., 9 P1.=P+5+J1.,Q1.=Q+5+J1.,PS1.=J2., 10 LEER(P1+R*COS(PSI),Q1+R*SIN(PSI))., 11 'FOR'S.=1'STEP'1'UNTIL'4'DO' 12 'BEGIN'PSI.=PSI+PIHALB., 13 LINE(P1+R*COS(PSI),Q1+R*SIN(PSI))., 14 'END".,I.=I+1 15 'END'QUAD., 16 R.=5*1.4142., 17 PIHALB.=3.14159*.5.,P14T.=PIHALB*.5., 18 I.=0., 19 SERIE(10.0,10.0,22,12,QUAD) 20 'END' SCHOTTER., 1 'REAL'P,Q,P1,Q1,XM,YM,HOR,VER,JLI,JRE,JUN,JOB., 5 'INTEGER'I,M,M,T., 7 'PROCEDURE'SERIE(QUER,HOCH,XMAL,YMAL,FIGUR)., 8 'VALUE'QUER,HOCH,XMAL,YMAL., 9 'REAL'QUER,HOCH., 10 'INTEGER'XMAL,YMAL., 11 'PROCEDURE'FIGUR., 12 'BEGIN' 13 'REAL'YANF., 14 'INTEGER'COUNTX,COUNTY., 15 P.=-QUER*XMAL*.5., 16 Q.=YANF.=-HOCH*YMAL*.5., 17 'FOR'COUNTX.=1'STEP'1'UNTIL'XMAL'DO' 18 'BEGIN'Q.=YANF., 19 'FOR'COUNTY.=1'STEP'1'UNTIL'YMAL'DO' 20 'BEGIN'FIGUR.,Q.=Q+HOCH 21 'END'.,P.=P+QUER 22 'END'., 23 LEER(-148.0,-105.0).,CLOSE., 24 SONK(11)., 25 OPBEN(X,Y) 26 'END'SERIE.,
gravel
What is missing in this ALGOL code are the seeds necessary for the reproduction of the plot. The author went down a rabbit hole to investigate and calculate different values, but managed to determine them to be “(1922110153) for the x and y shift seed, and (1769133315) for the rotation seed”. They also provided a Python translation
import math
import drawsvg as draw
class Random:
def __init__(self, seed):
self.JI = seed
def next(self, JA, JE):
self.JI = (self.JI * 5) % 2147483648
return self.JI / 2147483648 * (JE-JA) + JA
def draw_square(g, x, y, i, r1, r2):
r = 5 * 1.4142
pi = 3.14159
move_limit = 5 * i / 264
twist_limit = pi/4 * i / 264
y_center = y + 5 + r1.next(-move_limit, move_limit)
x_center = x + 5 + r1.next(-move_limit, move_limit)
angle = r2.next(pi/4 - twist_limit, pi/4 + twist_limit)
p = draw.Path()
p.M(x_center + r * math.sin(angle), y_center + r * math.cos(angle))
for step in range(4):
angle += pi / 2
p.L(x_center + r * math.sin(angle), y_center + r * math.cos(angle))
g.append(p)
def draw_plot(x_size, y_size, x_count, y_count, s1, s2):
r1 = Random(s1)
r2 = Random(s2)
d = draw.Drawing(180, 280, origin='center', style="background-color:#eae6e2")
g = draw.Group(stroke='#41403a', stroke_width='0.4', fill='none',
stroke_linecap="round", stroke_linejoin="round")
y = -y_size * y_count * 0.5
x0 = -x_size * x_count * 0.5
i = 0
for _ in range(y_count):
x = x0
for _ in range(x_count):
draw_square(g, x, y, i, r1, r2)
x += x_size
i += 1
y += y_size
d.append(g)
return d
d = draw_plot(10.0, 10.0, 12, 22, 1922110153, 1769133315).set_render_size(w=500)
print(d.as_svg())
I wanted to see if I could also translate this into R – base plot I can draw line segments very well, and I was curious to color the squares in my own way.
Most of this code translates simply, except that the “randomness” is actually a sequence of values, starting with a specific seed. I recently talked about an old article of mine that (abusively) uses the set.seed() function to generate specific “random” words
printStr <- function(str) paste(str, collapse="") set.seed(2505587); x <- sample(LETTERS, 5, replace=TRUE) set.seed(11135560);y <- sample(LETTERS, 5, replace=TRUE) paste(printStr(x), printStr(y)) ## [1] "HELLO WORLD"
which I wanted to revisit based on an article by Andrew Heiss.
THE Random The class in this Python translation produces an iterator that returns a “next” value each time it is called with a specific “seed” and two values
class Random:
def __init__(self, seed):
self.JI = seed
def next(self, JA, JE):
self.JI = (self.JI * 5) % 2147483648
return self.JI / 2147483648 * (JE-JA) + JA
r = Random(1)
r.next(2, 3)
## 2.0000000023283064
r.next(2, 3)
## 2.000000011641532
r.next(2, 3)
## 2.000000058207661
with the added complexity that subsequent calls update the seed itself.
When I first saw this, my mind went back to reading the “original” R paper “R: A Language for Data Analysis and Graphics” by Ross Ihaka and Robert Gentleman, in which I remember seeing the cool example of an OO system maintaining (non-global) state via <<-
Maintain the status of the total balance internal to the function
With this same trick, we can write an equivalent of Random class that also updates the seed internally
random <- function(seed)
list(
nextval = function(a, b)
seed <<- (seed * 5) %% 2147483648
seed / 2147483648 * (b-a) + a
)
r <- random(1)
print(r$nextval(2, 3), digits = 16)
## [1] 2.000000002328306
print(r$nextval(2, 3), digits = 16)
## [1] 2.000000011641532
print(r$nextval(2, 3), digits = 16)
## [1] 2.000000058207661
Cool!
The rest of the translation mostly aligns with the base plot syntax.
This is what I ended up with
draw_square <- function(x, y, i, r1, r2, col)
r = 5 * 1.4142
move_limit = 5 * i / 264
twist_limit = pi/4 * i / 264
y_center = y + 5 + r1$nextval(-move_limit, move_limit)
x_center = x + 5 + r1$nextval(-move_limit, move_limit)
angle = r2$nextval(pi/4 - twist_limit, pi/4 + twist_limit)
x0 <- x_center + r * sin(angle)
y0 <- y_center + r * cos(angle)
for (step in 1:4)
angle <- angle + pi / 2
x1 <- x_center + r * sin(angle)
y1 <- y_center + r * cos(angle)
segments(x0, y0, x1, y1, lwd = 1.75, col = col)
x0 <- x1
y0 <- y1
draw_plot <- function(x_size, y_size, x_count, y_count, s1, s2)
r1 = random(s1)
r2 = random(s2)
plot(NULL, NULL, xlim = c(-60, 60), ylim = c(120, -120), axes = FALSE, ann = FALSE)
y = -y_size * y_count * 0.5
x0 = -x_size * x_count * 0.5
i = 0
for (z in 1:y_count)
x = x0
for (zz in 1:x_count)
draw_square(x, y, i, r1, r2, "black")
x <- x + x_size
i <- i + 1
y <- y + y_size
draw_plot(10.0, 10.0, 12, 22, 1922110153, 1769133315)
Figure 1: “Gravel” in R
Which uses the special seeds discovered in this original post. Checking the rotations, this does indeed appear to match the original art.
But why stop there? Now that I can trace it, I can change things…what if I used a different set of seeds, for example swapping them?
draw_plot(10.0, 10.0, 12, 22, 1769133315, 1922110153)
Figure 2: ‘Schotter’ in R with swapped seeds
or completely different values?
draw_plot(10.0, 10.0, 12, 22, 12345, 67890)
Figure 3: ‘Schotter’ in R with new seeds
What if we changed the colors? I could plot the color based on the progress on the grid, which seems pretty cool to me.
draw_plot <- function(x_size, y_size, x_count, y_count, s1, s2)
r1 = random(s1)
r2 = random(s2)
plot(NULL, NULL, xlim = c(-60, 60), ylim = c(120, -120), axes = FALSE, ann = FALSE)
y = -y_size * y_count * 0.5
x0 = -x_size * x_count * 0.5
i = 0
for (z in 1:y_count)
x = x0
rcol <- scales::viridis_pal(option = "viridis")(y_count)[z]
for (zz in 1:x_count)
draw_square(x, y, i, r1, r2, rcol)
x <- x + x_size
i <- i + 1
y <- y + y_size
draw_plot(10.0, 10.0, 12, 22, 1922110153, 1769133315)
Figure 4: ‘Schotter’ in R with viridis colors
Since writing this article, I have seen other examples of similar work. This tool demonstrated a simplified version
suppressPackageStartupMessages(library(tidyverse))
crossing(x=0:10, y=x) |>
mutate(dx = rnorm(n(), 0, (y/20)^1.5),
dy = rnorm(n(), 0, (y/20)^1.5)) |>
ggplot() +
geom_tile(aes(x=x+dx, y=y+dy, fill=y), colour='black',
lwd=2, width=1, height=1, alpha=0.8, show.legend=FALSE) +
scale_fill_gradient(high='#9f025e', low='#f9c929') +
scale_y_reverse() + theme_void()
while this one showed a book ‘Crisis Engineering’ with a similar idea
Cover of “Crisis Engineering”
I’m sure I’ve seen others too.
It was a fun exploration of artistically inspired code translation, and I got to stretch my “maintaining internal state” muscles a bit. I have no doubt that someone more artistic than me could do much more.
As always, I can be found on Mastodon and in the comments section below.
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