Math art

Create mathematical art with R

This package provides functions and data for creating mathematical art.

Note: Previously this package contained functions for generating data from parametric equations discovered by the mathematical artist Hamid Naderi Yeganeh. The equations, which are publically available, generate data that, when plotted without further processing, closely resemble Hamid's artworks. To ensure that I do not inadvertently receive credit for Hamid's work and out of respect for the artist I have removed any functions and plots derived from his equations, and apologise to Hamid for any confusion this may have caused.

How to use

Install the packages

install.packages(c("devtools", "mapproj", "tidyverse", "ggforce", "Rcpp"))
devtools::install_github("marcusvolz/mathart")

Load the libraries

library(mathart)
library(ggforce)
library(Rcpp)
library(tidyverse)

Create mollusc shells

The shell model is described in the paper "Models for mollusc shell shape" by M.B. Cortie (1989).

Create harmonographs

Refer to the Wikipedia article for details about harmonographs.

df1 <- harmonograph(A1 = 1, A2 = 1, A3 = 1, A4 = 1,
                    d1 = 0.004, d2 = 0.0065, d3 = 0.008, d4 = 0.019,
                    f1 = 3.001, f2 = 2, f3 = 3, f4 = 2,
                    p1 = 0, p2 = 0, p3 = pi/2, p4 = 3*pi/2) %>% mutate(id = 1)

df2 <- harmonograph(A1 = 1, A2 = 1, A3 = 1, A4 = 1,
                    d1 = 0.0085, d2 = 0, d3 = 0.065, d4 = 0,
                    f1 = 2.01, f2 = 3, f3 = 3, f4 = 2,
                    p1 = 0, p2 = 7*pi/16, p3 = 0, p4 = 0) %>% mutate(id = 2)

df3 <- harmonograph(A1 = 1, A2 = 1, A3 = 1, A4 = 1,
                    d1 = 0.039, d2 = 0.006, d3 = 0, d4 = 0.0045,
                    f1 = 10, f2 = 3, f3 = 1, f4 = 2,
                    p1 = 0, p2 = 0, p3 = pi/2, p4 = 0) %>% mutate(id = 3)

df4 <- harmonograph(A1 = 1, A2 = 1, A3 = 1, A4 = 1,
                    d1 = 0.02, d2 = 0.0315, d3 = 0.02, d4 = 0.02,
                    f1 = 2, f2 = 6, f3 = 1.002, f4 = 3,
                    p1 = pi/16, p2 = 3*pi/2, p3 = 13*pi/16, p4 = pi) %>% mutate(id = 4)

df <- rbind(df1, df2, df3, df4)

p <- ggplot() +
  geom_path(aes(x, y), df, alpha = 0.25, size = 0.5) +
  coord_equal() +
  facet_wrap(~id, nrow = 2) +
  theme_blankcanvas(margin_cm = 0)

ggsave("harmonograph01.png", p, width = 20, height = 20, units = "cm")

Create Lissajous curves

Refer to the Wikipedia article for details about Lissajous curves.

set.seed(1)

df <- 1:100 %>% map_df(~lissajous(a = runif(1, 0, 10), A = runif(1, 0, 1)), .id = "id")

p <- ggplot() +
  geom_path(aes(x, y), df, size = 0.25, lineend = "round") +
  facet_wrap(~id, nrow = 10) +
  coord_equal() +
  theme_blankcanvas(margin_cm = 1)

ggsave("lissajous001.png", p, width = 20, height = 20, units = "cm", dpi = 300)

Lissajous curves as k-nearest neighbour graphs

Refer to the Wikipedia article for details about k-nearest neighbors graphs.

set.seed(2)
df <- lissajous(a = runif(1, 0, 2), b = runif(1, 0, 2), A = runif(1, 0, 2), B = runif(1, 0, 2), d = 200) %>%
  sample_n(1001) %>%
  k_nearest_neighbour_graph(40)

p <- ggplot() +
  geom_segment(aes(x, y, xend = xend, yend = yend), df, size = 0.03) +
  coord_equal() +
  theme_blankcanvas(margin_cm = 0)

ggsave("knn_lissajous_002.png", p, width = 25, height = 25, units = "cm")

Rose curves

Refer to the Wikipedia article for details about rose curves.

df <- data.frame(x = numeric(0), y = numeric(0), n = integer(0), d = integer(0))

for(n in 1:10) {
  for(d in 1:10) {
    result <- rose_curve(n, d) %>% mutate(n = n, d = d)
    df <- rbind(df, result)
  }
}

p <- ggplot() +
  geom_path(aes(x, y), df, size = 0.35, lineend = "round") +
  facet_grid(d ~ n) +
  coord_equal() +
  theme_blankcanvas(margin_cm = 1)

ggsave("rose_curves.png", p, width = 20, height = 20, units = "cm")

Lorenz attractor

The Rcpp implementations for the attractor functions in this package are from the blog post Drawing 10 Million Points With ggplot by Antonio Sanchez Chinchon.

df <- lorenz_attractor(a=20, b=8/3, c=28, n=1000000)

p <- ggplot() +
  geom_path(aes(x, z), df, alpha = 0.15, size = 0.03) +
  coord_equal() +
  xlim(-25, 25) + ylim(2.5, 52.5) +
  theme_blankcanvas(margin_cm = 0)

ggsave("lorenz_attractor.png", p, width = 20, height = 20, units = "cm")

Rapidly exploring random tree

Refer to the Wikipedia article for details about rapidly exploring random trees.

# Generate rrt edges
set.seed(1)
df <- rapidly_exploring_random_tree() %>% mutate(id = 1:nrow(.))

# Create plot
p <- ggplot() +
  geom_segment(aes(x, y, xend = xend, yend = yend, size = -id, alpha = -id), df, lineend = "round") +
  coord_equal() +
  scale_size_continuous(range = c(0.1, 0.75)) +
  scale_alpha_continuous(range = c(0.1, 1)) +
  theme_blankcanvas(margin_cm = 0)

# Save plot
ggsave("rapidly_exploring_random_tree.png", p, width = 20, height = 20, units = "cm")

Fractal ferns

Refer to the Wikipedia article for details about fractal ferns.

n <- 250000

params1 <- data.frame(
  a <- c(0, 0.85, 0.2, -0.15),
  b <- c(0, 0.04, -0.26, 0.28),
  c <- c(0, -0.04, 0.23, 0.26),
  d <- c(0.16, 0.85, 0.22, 0.24),
  e <- c(0, 0, 0, 0),
  f <- c(0, 1.6, 1.6, 0.44),
  p <- c(0.01, 0.85, 0.07, 0.07)
)

params2 <- data.frame(
  a <- c(0, 0.85, 0.09, -0.09),
  b <- c(0, 0.02, -0.28, 0.28),
  c <- c(0, -0.02, 0.3, 0.3),
  d <- c(0.25, 0.83, 0.11, 0.09),
  e <- c(0, 0, 0, 0),
  f <- c(-0.14, 1, 0.6, 0.7),
  p <- c(0.02, 0.84, 0.07, 0.07)
)

df1 <- fractal_fern(n = n, a = params1$a, b = params1$b, c_ = params1$c, d = params1$d, e = params1$e,
                   f = params1$f, p = params1$p) %>% mutate(id = 1)

df2 <- fractal_fern(n = n, a = params2$a, b = params2$b, c_ = params2$c, d = params2$d, e = params2$e,
                    f = params2$f, p = params2$p) %>% mutate(id = 2)

df <- rbind(df1, df2 %>% mutate(x = x*1.75, y = y*1.75))

p <- ggplot() +
  geom_point(aes(x, y), df, size = 0.03, alpha = 0.06) +
  coord_equal() +
  facet_wrap(~id, nrow = 1) +
  theme_blankcanvas(margin_cm = 1)

ggsave("fern01.png", width = 20, height = 20, units = "cm")

k-d tree

Refer to the Wikipedia article for details about k-d trees.

points <- mathart::points

result <- kdtree(points)

p <- ggplot() +
  geom_segment(aes(x, y, xend = xend, yend = yend), result) +
  coord_equal() +
  xlim(0, 10000) + ylim(0, 10000) +
  theme_blankcanvas(margin_cm = 0)

ggsave("kdtree.png", p, width = 20, height = 20, units = "in")

Weiszfeld convergence

This plot shows iterations of Weiszfeld's algorithm for finding the geometric median of a given set of points. The algorithm is initialised from 10,000 locations. Refer to the Wikipedia article for details about the geometric median and Weiszfeld's algorithm.

set.seed(1)

terminals <- data.frame(x = runif(10, 0, 10000), y = runif(10, 0, 10000))

df <- 1:10000 %>%
  map_df(~weiszfeld(terminals, c(points$x[.], points$y[.])), .id = "id")

p <- ggplot() +
  geom_point(aes(x, y), points, size = 1, alpha = 0.25) +
  geom_point(aes(x, y), terminals, size = 5, alpha = 1) +
  geom_line(aes(x, y, group = id), df, colour = "black", size = 0.5, alpha = 0.03) +
  coord_equal() +
  xlim(0, 10000) +
  ylim(0, 10000) +
  theme_blankcanvas(margin_cm = 0)

ggsave("weiszfeld.png", p, width = 20, height = 20, units = "in")