In WilliamTylerBradley/flowfieldfigments: Flow Field Generative Art

knitr::opts_chunk$set(
  collapse = TRUE,
  comment = "#>"
)
options(rmarkdown.html_vignette.check_title = FALSE)

Background

Each output from this package starts with a set of anchor points where the paths begin. This vignette displays the possible variation in anchor points. There are three styles, random, spiral, and diamond. The number of points ranges uniformly from 50 to 1,500. The paths' starting direction and length determine the color and are generated by simplex noise. The following code examines these pieces individually and puts them together for comparison.

Layout

The first layout style is random. random yields points from a multivariate normal distribution. The standard deviation is the number of points, size, times four divided by the number of digits in size. So for a size of 50 to under 100, the standard deviation is double the size. While for 1,000 and up, it's just size. This modification tends to balance the images out nicely over a range of values.

The following images display the random anchor point style for a middle size value of 750 and the extremes of 50 and 1,500.

library(flowfieldfigments)
library(tidyverse)
library(scales)
library(mvtnorm)
library(ambient)

# anchor_layout == "random"
set.seed(1)
size <- 750

points <- as.data.frame(mvtnorm::rmvnorm(
  n = size,
  sigma = diag(size * 4 / (floor(log10(size)) + 1),
    nrow = 2
  )
)) %>%
  rename(
    x = V1,
    y = V2
  ) %>%
  mutate(id = row_number())

axes_limits <- max(c(abs(c(
  points$x,
  points$y
))))

ggplot(
  data = points,
  aes(x, y)
) +
  geom_point(size = .5) +
  scale_x_continuous(limits = c(-axes_limits, axes_limits)) +
  scale_y_continuous(limits = c(-axes_limits, axes_limits)) +
  coord_equal() +
  theme_void()

:::: {style="display: grid; grid-template-columns: 1fr 1fr; align-items: center; justify-items: center; grid-column-gap: 1px;"} ::: {}

set.seed(1)
size <- 50

points <- as.data.frame(mvtnorm::rmvnorm(
  n = size,
  sigma = diag(size * 4 / (floor(log10(size)) + 1),
    nrow = 2
  )
)) %>%
  rename(
    x = V1,
    y = V2
  ) %>%
  mutate(id = row_number())

axes_limits <- max(c(abs(c(
  points$x,
  points$y
))))

ggplot(
  data = points,
  aes(x, y)
) +
  geom_point(size = .5) +
  scale_x_continuous(limits = c(-axes_limits, axes_limits)) +
  scale_y_continuous(limits = c(-axes_limits, axes_limits)) +
  coord_equal() +
  theme_void()

::: ::: {}

set.seed(1)
size <- 1500

points <- as.data.frame(mvtnorm::rmvnorm(
  n = size,
  sigma = diag(size * 4 / (floor(log10(size)) + 1),
    nrow = 2
  )
)) %>%
  rename(
    x = V1,
    y = V2
  ) %>%
  mutate(id = row_number())

axes_limits <- max(c(abs(c(
  points$x,
  points$y
))))

ggplot(
  data = points,
  aes(x, y)
) +
  geom_point(size = .5) +
  scale_x_continuous(limits = c(-axes_limits, axes_limits)) +
  scale_y_continuous(limits = c(-axes_limits, axes_limits)) +
  coord_equal() +
  theme_void()

::: ::::

spiral is the next style. This one follows a spiral based on the golden ratio. Because the output will take up the same space, smaller size values will appear more open while larger size values are denser. Unfortunately, sometimes there is a little hole in the middle depending on cutoffs based on size. However, this is a better result than overcrowding which can happen if there is always a point in the direct center.

# anchor_layout == "spiral"
size <- 750

golden <- ((sqrt(5) + 1) / 2) * (2 * pi)

points <- data.frame(
  x = sqrt(seq(1, size)) * cos(golden * seq(1, size)) * 2.5,
  y = sqrt(seq(1, size)) * sin(golden * seq(1, size)) * 2.5
) %>%
  mutate(id = row_number())

axes_limits <- max(c(abs(c(
  points$x,
  points$y
))))

ggplot(
  data = points,
  aes(x, y)
) +
  geom_point(size = .5) +
  scale_x_continuous(limits = c(-axes_limits, axes_limits)) +
  scale_y_continuous(limits = c(-axes_limits, axes_limits)) +
  coord_equal() +
  theme_void()

:::: {style="display: grid; grid-template-columns: 1fr 1fr; align-items: center; justify-items: center; grid-column-gap: 1px;"} ::: {}

size <- 50

golden <- ((sqrt(5) + 1) / 2) * (2 * pi)

points <- data.frame(
  x = sqrt(seq(1, size)) * cos(golden * seq(1, size)) * 2.5,
  y = sqrt(seq(1, size)) * sin(golden * seq(1, size)) * 2.5
) %>%
  mutate(id = row_number())

axes_limits <- max(c(abs(c(
  points$x,
  points$y
))))

ggplot(
  data = points,
  aes(x, y)
) +
  geom_point(size = .5) +
  scale_x_continuous(limits = c(-axes_limits, axes_limits)) +
  scale_y_continuous(limits = c(-axes_limits, axes_limits)) +
  coord_equal() +
  theme_void()

::: ::: {}

size <- 1500

golden <- ((sqrt(5) + 1) / 2) * (2 * pi)

points <- data.frame(
  x = sqrt(seq(1, size)) * cos(golden * seq(1, size)) * 2.5,
  y = sqrt(seq(1, size)) * sin(golden * seq(1, size)) * 2.5
) %>%
  mutate(id = row_number())

axes_limits <- max(c(abs(c(
  points$x,
  points$y
))))

ggplot(
  data = points,
  aes(x, y)
) +
  geom_point(size = .5) +
  scale_x_continuous(limits = c(-axes_limits, axes_limits)) +
  scale_y_continuous(limits = c(-axes_limits, axes_limits)) +
  coord_equal() +
  theme_void()

::: ::::

The final layout is diamond. This is a tilted grid with a width of size's square root. Because the paths' movements can follow along diagonals, this layout can produce rows of straight lines.

# anchor_layout == "diamond"
size <- 750

grid_width <- ifelse(size <= 1500 / 2, ceiling(sqrt(size)), floor(sqrt(size)))

points <- expand_grid(
  x_start = seq(1, grid_width) - (grid_width / 2) - .5,
  y_start = seq(1, grid_width) - (grid_width / 2) - .5
) %>%
  mutate(
    x = (x_start * cos(45 * pi / 180) - y_start * sin(45 * pi / 180)) * 5,
    y = (x_start * sin(45 * pi / 180) + y_start * cos(45 * pi / 180)) * 5,
    id = row_number()
  ) %>%
  select(-x_start, -y_start)

axes_limits <- max(c(abs(c(
  points$x,
  points$y
))))

ggplot(
  data = points,
  aes(x, y)
) +
  geom_point(size = .5) +
  scale_x_continuous(limits = c(-axes_limits, axes_limits)) +
  scale_y_continuous(limits = c(-axes_limits, axes_limits)) +
  coord_equal() +
  theme_void()

:::: {style="display: grid; grid-template-columns: 1fr 1fr; align-items: center; justify-items: center; grid-column-gap: 1px;"} ::: {}

size <- 50

grid_width <- ifelse(size <= 1500 / 2, ceiling(sqrt(size)), floor(sqrt(size)))

points <- expand_grid(
  x_start = seq(1, grid_width) - (grid_width / 2) - .5,
  y_start = seq(1, grid_width) - (grid_width / 2) - .5
) %>%
  mutate(
    x = (x_start * cos(45 * pi / 180) - y_start * sin(45 * pi / 180)) * 5,
    y = (x_start * sin(45 * pi / 180) + y_start * cos(45 * pi / 180)) * 5,
    id = row_number()
  ) %>%
  select(-x_start, -y_start)

axes_limits <- max(c(abs(c(
  points$x,
  points$y
))))

ggplot(
  data = points,
  aes(x, y)
) +
  geom_point(size = .5) +
  scale_x_continuous(limits = c(-axes_limits, axes_limits)) +
  scale_y_continuous(limits = c(-axes_limits, axes_limits)) +
  coord_equal() +
  theme_void()

::: ::: {}

size <- 1500

grid_width <- ifelse(size <= 1500 / 2, ceiling(sqrt(size)), floor(sqrt(size)))

points <- expand_grid(
  x_start = seq(1, grid_width) - (grid_width / 2) - .5,
  y_start = seq(1, grid_width) - (grid_width / 2) - .5
) %>%
  mutate(
    x = (x_start * cos(45 * pi / 180) - y_start * sin(45 * pi / 180)) * 5,
    y = (x_start * sin(45 * pi / 180) + y_start * cos(45 * pi / 180)) * 5,
    id = row_number()
  ) %>%
  select(-x_start, -y_start)

axes_limits <- max(c(abs(c(
  points$x,
  points$y
))))

ggplot(
  data = points,
  aes(x, y)
) +
  geom_point(size = .5) +
  scale_x_continuous(limits = c(-axes_limits, axes_limits)) +
  scale_y_continuous(limits = c(-axes_limits, axes_limits)) +
  coord_equal() +
  theme_void()

::: ::::

Color

The color section discusses the paths' starting direction, as an angle, and length, documented as distance and percentage. These values specify the final color. Because these values get generated before the paths, they can be thought of as points' attributes.

The initial angle for the paths determines the hue. The following code example starts with two directional values (x_direction and y_direction), standardizes them to a unit vector, and finally uses atan2 to get the angle. This example sets up for later code when the directional vectors are drawn from noise, but the rest of the procedure is the same.

There is a randomly generated value for hue_turn that rotates the colors. This way, the colors can end up pointing in any direction. For these examples, this variable is set to zero, so they all follow the traditional hue values.

seeds <- sample(1:10000, 3)
hue_turn <- 0

angle_data <- expand_grid(
  x_direction = seq(-1, 1, by = .5),
  y_direction = seq(-1, 1, by = .5)
) %>%
  mutate(vector_length = sqrt(x_direction^2 + y_direction^2)) %>%
  mutate(
    x_direction = x_direction / vector_length,
    y_direction = y_direction / vector_length
  ) %>%
  select(-vector_length) %>%
  mutate(angle = (atan2(y_direction, x_direction) * 180 / pi) %% 360 + hue_turn) %>%
  filter(!is.na(angle)) %>%
  distinct() %>%
  mutate(
    x = cos(angle * pi / 180),
    y = sin(angle * pi / 180),
    label_text = paste0(
      "x_direction = ", round(x_direction, 3),
      ",\n y_direction = ", round(y_direction, 3)
    ),
    hjust = if_else(x >= 0, -.15, 1.15),
    vjust = if_else(y >= 0, -.05, 1.05),
    text_angle = if_else(x >= 0, angle, angle + 180)
  ) %>%
  mutate(vjust = if_else(angle %in% c(0, 90, 180, 270), .5, vjust))

# basically just running around the unit circle
ggplot(
  data = angle_data,
  aes(
    x = x,
    y = y,
    color = angle,
    label = label_text,
    hjust = hjust,
    vjust = vjust,
    angle = text_angle
  )
) +
  geom_point() +
  geom_text(size = 1.25) +
  scale_x_continuous(limits = c(-2.5, 2.5)) +
  scale_y_continuous(limits = c(-2.5, 2.5)) +
  coord_equal() +
  theme(
    text = element_text(size = 7.5),
    legend.key.size = unit(.25, "cm")
  )

The distance value determines the chroma. The code will need a parameter from 0 to 100, but the data mostly range symmetrically around 0, possibly from -2 to 2. Using the standard deviation of the data, the pnorm function can convert these values to be between 0 and 1. Then multiplying by 100 spreads it across the acceptable range. Of course, different effects will come from different distributions, but this one works out nicely. (Also, it's not precisely the chroma value because the color scheme is at an angle and maxes out at 63. See the color scheme vignette.)

distance_data <- data.frame(
  distance_1 = rnorm(1000),
  distance_2 = runif(1000, -1, 1),
  distance_3 = c(rnorm(500), runif(500, -1, 1))
) %>%
  mutate(
    distance_1 = pnorm(distance_1, 0, sd = sd(distance_1)) * 100,
    distance_2 = pnorm(distance_2, 0, sd = sd(distance_2)) * 100,
    distance_3 = pnorm(distance_3, 0, sd = sd(distance_3)) * 100
  )

:::: {style="display: grid; grid-template-columns: 1fr 1fr 1fr; align-items: center; justify-items: center; grid-column-gap: 1px;"} ::: {}

ggplot(
  data = distance_data,
  aes(x = distance_1)
) +
  geom_histogram(bins = 25)

::: ::: {}

ggplot(
  data = distance_data,
  aes(x = distance_2)
) +
  geom_histogram(bins = 25)

::: ::: {}

ggplot(
  data = distance_data,
  aes(x = distance_3)
) +
  geom_histogram(bins = 25)

::: ::::

Now that the angle and distance examples are set, the actual code can be investigated. The get_vectors function will generate the paths from simplex noise and is used here for the initial angle. Notice the familiar code with x_direction and y_direction.

get_vectors <- function(points, seeds) {
  vectors <- points %>%
    mutate(
      x_direction = gen_simplex(x,
        y,
        frequency = .01,
        seed = seeds[1]
      ),
      y_direction = gen_simplex(x,
        y,
        frequency = .01,
        seed = seeds[2]
      )
    ) %>%
    mutate(vector_length = sqrt(x_direction^2 + y_direction^2)) %>%
    mutate(
      x_direction = x_direction / vector_length,
      y_direction = y_direction / vector_length
    ) %>%
    select(-vector_length)
}

We will use the diamond layout with a size of 750 to set the anchor points. The get_vectors function uses this dataset, generates the initial direction and distance, calculates the angle and percentage, then gets the color. We can pull the result apart in the following sections.

size <- 750

grid_width <- ifelse(size <= 1500 / 2, ceiling(sqrt(size)), floor(sqrt(size)))

points <- expand_grid(
  x_start = seq(1, grid_width) - (grid_width / 2) - .5,
  y_start = seq(1, grid_width) - (grid_width / 2) - .5
) %>%
  mutate(
    x = (x_start * cos(45 * pi / 180) - y_start * sin(45 * pi / 180)) * 5,
    y = (x_start * sin(45 * pi / 180) + y_start * cos(45 * pi / 180)) * 5,
    id = row_number()
  ) %>%
  select(-x_start, -y_start)

points <- get_vectors(points, seeds) %>%
  mutate(distance = gen_simplex(points$x,
    points$y,
    frequency = .01,
    seed = seeds[3]
  )) %>%
  mutate(
    angle = (atan2(y_direction, x_direction) * 180 / pi) %% 360 + hue_turn,
    percentage = pnorm(distance, mean = 0, sd = sd(distance)) * 100
  ) %>%
  rowwise() %>%
  mutate(hex_color = flowfieldfigments::get_color(angle, percentage)) %>%
  ungroup() %>%
  mutate(
    x_color = percentage * cos(angle * pi / 180),
    y_color = percentage * sin(angle * pi / 180),
    time = 0
  ) %>%
  filter(!is.na(hex_color))

axes_limits <- max(c(abs(c(
  points$x,
  points$y
))))

ggplot(
  data = points,
  aes(x, y, color = hex_color)
) +
  geom_point(size = .5) +
  scale_color_identity() +
  scale_x_continuous(limits = c(-axes_limits, axes_limits)) +
  scale_y_continuous(limits = c(-axes_limits, axes_limits)) +
  coord_equal() +
  theme_void()

From here, we can analyze the angle values. An angle of 135 is common, while values from 0 to 45 are not. The colors show this with a large green region and very little red.

:::: {style="display: grid; grid-template-columns: 1fr 1fr; align-items: center; justify-items: center; grid-column-gap: 1px;"} ::: {}

ggplot(
  data = points,
  aes(x = angle)
) +
  geom_histogram(bins = 36) +
  scale_x_continuous(
    limits = c(0, 360),
    oob = scales::oob_keep,
    breaks = seq(0, 360, 45)
  ) +
  coord_polar(
    direction = -1,
    start = 270 * pi / 180
  )

::: ::: {}

colors <- data.frame(angle = seq(0, 360, 30)) %>%
  rowwise() %>%
  mutate(color = 
           flowfieldfigments::get_color(
             angle, 100)) %>%
  pull(color)

ggplot(
  data = points,
  aes(x, y, color = angle)
) +
  geom_point() +
  scale_x_continuous(limits = c(-axes_limits, 
                                axes_limits)) +
  scale_y_continuous(limits = c(-axes_limits, 
                                axes_limits)) +
  coord_equal() +
  theme_void() +
  scale_colour_gradientn(
    colors = colors,
    breaks = seq(0, 360, 45)
  )

::: ::::

:::: {style="display: grid; grid-template-columns: 1fr 1fr; align-items: center; justify-items: center; grid-column-gap: 1px;"} ::: {}

ggplot(
  data = points,
  aes(x = angle)
) +
  geom_histogram(bins = 36) +
  scale_x_continuous(
    limits = c(0, 360),
    oob = scales::oob_keep,
    breaks = seq(0, 360, 45)
  ) +
  coord_polar(
    direction = -1,
    start = 270 * pi / 180
  )

::: ::: {}

colors <- data.frame(angle = seq(0, 360, 30)) %>%
  rowwise() %>%
  mutate(color = flowfieldfigments::get_color(angle, 100)) %>%
  pull(color)

ggplot(
  data = points,
  aes(x, y, color = angle)
) +
  geom_point() +
  scale_x_continuous(limits = c(-axes_limits, axes_limits)) +
  scale_y_continuous(limits = c(-axes_limits, axes_limits)) +
  coord_equal() +
  theme_void() +
  scale_colour_gradientn(
    colors = colors,
    breaks = seq(0, 360, 45)
  )

::: ::::

The distance values range from -1 to 1 for this pull. Notice that the highlights from the distance map do not match any specific region of the earlier angle map. These values are unconnected. The percentage values, however, are a function of the distance values. So, a strong match between the two maps exists.

:::: {style="display: grid; grid-template-columns: 1fr 1fr; align-items: center; justify-items: center; grid-column-gap: 1px;"} ::: {}

ggplot(
  data = points,
  aes(x = distance)
) +
  geom_histogram(bins = 25)

::: ::: {}

ggplot(
  data = points,
  aes(x, y, color = distance)
) +
  geom_point() +
  scale_x_continuous(limits = c(-axes_limits, 
                                axes_limits)) +
  scale_y_continuous(limits = c(-axes_limits, 
                                axes_limits)) +
  coord_equal() +
  theme_void()

::: ::::

:::: {style="display: grid; grid-template-columns: 1fr 1fr; align-items: center; justify-items: center; grid-column-gap: 1px;"} ::: {}

ggplot(
  data = points,
  aes(x = distance)
) +
  geom_histogram(bins = 25)

::: ::: {}

ggplot(
  data = points,
  aes(x, y, color = distance)
) +
  geom_point() +
  scale_x_continuous(limits = c(-axes_limits, 
                                axes_limits)) +
  scale_y_continuous(limits = c(-axes_limits, 
                                axes_limits)) +
  coord_equal() +
  theme_void()

::: ::::

:::: {style="display: grid; grid-template-columns: 1fr 1fr; align-items: center; justify-items: center; grid-column-gap: 1px;"} ::: {}

ggplot(
  data = points,
  aes(x = percentage)
) +
  geom_histogram(bins = 25)

::: ::: {}

ggplot(
  data = points,
  aes(x, y, color = percentage)
) +
  geom_point() +
  scale_x_continuous(limits = c(-axes_limits, 
                                axes_limits)) +
  scale_y_continuous(limits = c(-axes_limits, 
                                axes_limits)) +
  coord_equal() +
  theme_void()

::: ::::

:::: {style="display: grid; grid-template-columns: 1fr 1fr; align-items: center; justify-items: center; grid-column-gap: 1px;"} ::: {}

ggplot(
  data = points,
  aes(x = percentage)
) +
  geom_histogram(bins = 25)

::: ::: {}

ggplot(
  data = points,
  aes(x, y, color = percentage)
) +
  geom_point() +
  scale_x_continuous(limits = c(-axes_limits, 
                                axes_limits)) +
  scale_y_continuous(limits = c(-axes_limits, 
                                axes_limits)) +
  coord_equal() +
  theme_void()

::: ::::

Finally, we can plot both angle and percentage from the points to see where they fall on the color scheme. This plot shows that while there are a lot of points in the green hue, a lot of them have a low percentage value and end up gray. This result matches the plot of the point attributes.

circle <- data.frame(value = c(seq(0, 360, .5), 0)) %>%
  mutate(x = 100 * cos(value * pi/180),
         y = 100 * sin(value * pi/180))

ggplot() +
  geom_polygon(data = circle,
               aes(x, y),
               color = "black",
               fill = "NA") +
  geom_point(data = points,
             aes(x_color, y_color, color = hex_color)) +
  scale_color_identity() +
  coord_equal() +
  theme_void()

Results comparison

The final section runs the previous code for different anchor layouts and sizes. For each set, the rows have size of 50, 750, and 1,500. There's a trend of angles following along the diagonals or flowing in loops. There are not any noteworthy differences between layout styles.

Random

:::: {style="display: grid; grid-template-columns: 1fr 1fr 1fr; align-items: center; justify-items: center; grid-column-gap: 1px;"} ::: {}

set.seed(2)
seeds <- sample(1:10000, 3)

size <- 50

points <- as.data.frame(mvtnorm::rmvnorm(
  n = size,
  sigma = diag(size * 4 / (floor(log10(size)) + 1),
    nrow = 2
  )
)) %>%
  rename(
    x = V1,
    y = V2
  ) %>%
  mutate(id = row_number())

points <- get_vectors(points, seeds) %>%
  mutate(distance = gen_simplex(points$x,
                                points$y,
                                frequency = .01,
                                seed = seeds[3])) %>%
  mutate(angle = (atan2(y_direction, x_direction) * 180/pi) %% 360 + hue_turn,
         percentage = pnorm(distance, mean = 0, sd = sd(distance)) * 100) %>%
  rowwise() %>%
  mutate(hex_color = flowfieldfigments::get_color(angle, percentage)) %>%
  ungroup() %>%
  mutate(x_color = percentage * cos(angle * pi/180),
         y_color = percentage * sin(angle * pi/180),
         time = 0) %>%
  filter(!is.na(hex_color))

ggplot() +
  geom_polygon(data = circle,
               aes(x, y),
               color = "black",
               fill = "NA") +
  geom_point(data = points,
             aes(x_color, y_color, color = hex_color)) +
  scale_color_identity() +
  coord_equal() +
  theme_void()

::: ::: {}

set.seed(3)
seeds <- sample(1:10000, 3)

size <- 50

points <- as.data.frame(mvtnorm::rmvnorm(
  n = size,
  sigma = diag(size * 4 / (floor(log10(size)) + 1),
    nrow = 2
  )
)) %>%
  rename(
    x = V1,
    y = V2
  ) %>%
  mutate(id = row_number())

points <- get_vectors(points, seeds) %>%
  mutate(distance = gen_simplex(points$x,
                                points$y,
                                frequency = .01,
                                seed = seeds[3])) %>%
  mutate(angle = (atan2(y_direction, x_direction) * 180/pi) %% 360 + hue_turn,
         percentage = pnorm(distance, mean = 0, sd = sd(distance)) * 100) %>%
  rowwise() %>%
  mutate(hex_color = flowfieldfigments::get_color(angle, percentage)) %>%
  ungroup() %>%
  mutate(x_color = percentage * cos(angle * pi/180),
         y_color = percentage * sin(angle * pi/180),
         time = 0) %>%
  filter(!is.na(hex_color))

ggplot() +
  geom_polygon(data = circle,
               aes(x, y),
               color = "black",
               fill = "NA") +
  geom_point(data = points,
             aes(x_color, y_color, color = hex_color)) +
  scale_color_identity() +
  coord_equal() +
  theme_void()

::: ::: {}

set.seed(4)
seeds <- sample(1:10000, 3)

size <- 50

points <- as.data.frame(mvtnorm::rmvnorm(
  n = size,
  sigma = diag(size * 4 / (floor(log10(size)) + 1),
    nrow = 2
  )
)) %>%
  rename(
    x = V1,
    y = V2
  ) %>%
  mutate(id = row_number())

points <- get_vectors(points, seeds) %>%
  mutate(distance = gen_simplex(points$x,
                                points$y,
                                frequency = .01,
                                seed = seeds[3])) %>%
  mutate(angle = (atan2(y_direction, x_direction) * 180/pi) %% 360 + hue_turn,
         percentage = pnorm(distance, mean = 0, sd = sd(distance)) * 100) %>%
  rowwise() %>%
  mutate(hex_color = flowfieldfigments::get_color(angle, percentage)) %>%
  ungroup() %>%
  mutate(x_color = percentage * cos(angle * pi/180),
         y_color = percentage * sin(angle * pi/180),
         time = 0) %>%
  filter(!is.na(hex_color))

ggplot() +
  geom_polygon(data = circle,
               aes(x, y),
               color = "black",
               fill = "NA") +
  geom_point(data = points,
             aes(x_color, y_color, color = hex_color)) +
  scale_color_identity() +
  coord_equal() +
  theme_void()

::: ::: {}

set.seed(5)
seeds <- sample(1:10000, 3)

size <- 750

points <- as.data.frame(mvtnorm::rmvnorm(
  n = size,
  sigma = diag(size * 4 / (floor(log10(size)) + 1),
    nrow = 2
  )
)) %>%
  rename(
    x = V1,
    y = V2
  ) %>%
  mutate(id = row_number())

points <- get_vectors(points, seeds) %>%
  mutate(distance = gen_simplex(points$x,
                                points$y,
                                frequency = .01,
                                seed = seeds[3])) %>%
  mutate(angle = (atan2(y_direction, x_direction) * 180/pi) %% 360 + hue_turn,
         percentage = pnorm(distance, mean = 0, sd = sd(distance)) * 100) %>%
  rowwise() %>%
  mutate(hex_color = flowfieldfigments::get_color(angle, percentage)) %>%
  ungroup() %>%
  mutate(x_color = percentage * cos(angle * pi/180),
         y_color = percentage * sin(angle * pi/180),
         time = 0) %>%
  filter(!is.na(hex_color))

ggplot() +
  geom_polygon(data = circle,
               aes(x, y),
               color = "black",
               fill = "NA") +
  geom_point(data = points,
             aes(x_color, y_color, color = hex_color)) +
  scale_color_identity() +
  coord_equal() +
  theme_void()

::: ::: {}

set.seed(6)
seeds <- sample(1:10000, 3)

size <- 750

points <- as.data.frame(mvtnorm::rmvnorm(
  n = size,
  sigma = diag(size * 4 / (floor(log10(size)) + 1),
    nrow = 2
  )
)) %>%
  rename(
    x = V1,
    y = V2
  ) %>%
  mutate(id = row_number())

points <- get_vectors(points, seeds) %>%
  mutate(distance = gen_simplex(points$x,
                                points$y,
                                frequency = .01,
                                seed = seeds[3])) %>%
  mutate(angle = (atan2(y_direction, x_direction) * 180/pi) %% 360 + hue_turn,
         percentage = pnorm(distance, mean = 0, sd = sd(distance)) * 100) %>%
  rowwise() %>%
  mutate(hex_color = flowfieldfigments::get_color(angle, percentage)) %>%
  ungroup() %>%
  mutate(x_color = percentage * cos(angle * pi/180),
         y_color = percentage * sin(angle * pi/180),
         time = 0) %>%
  filter(!is.na(hex_color))

ggplot() +
  geom_polygon(data = circle,
               aes(x, y),
               color = "black",
               fill = "NA") +
  geom_point(data = points,
             aes(x_color, y_color, color = hex_color)) +
  scale_color_identity() +
  coord_equal() +
  theme_void()

::: ::: {}

set.seed(7)
seeds <- sample(1:10000, 3)

size <- 750

points <- as.data.frame(mvtnorm::rmvnorm(
  n = size,
  sigma = diag(size * 4 / (floor(log10(size)) + 1),
    nrow = 2
  )
)) %>%
  rename(
    x = V1,
    y = V2
  ) %>%
  mutate(id = row_number())

points <- get_vectors(points, seeds) %>%
  mutate(distance = gen_simplex(points$x,
                                points$y,
                                frequency = .01,
                                seed = seeds[3])) %>%
  mutate(angle = (atan2(y_direction, x_direction) * 180/pi) %% 360 + hue_turn,
         percentage = pnorm(distance, mean = 0, sd = sd(distance)) * 100) %>%
  rowwise() %>%
  mutate(hex_color = flowfieldfigments::get_color(angle, percentage)) %>%
  ungroup() %>%
  mutate(x_color = percentage * cos(angle * pi/180),
         y_color = percentage * sin(angle * pi/180),
         time = 0) %>%
  filter(!is.na(hex_color))

ggplot() +
  geom_polygon(data = circle,
               aes(x, y),
               color = "black",
               fill = "NA") +
  geom_point(data = points,
             aes(x_color, y_color, color = hex_color)) +
  scale_color_identity() +
  coord_equal() +
  theme_void()

::: ::: {}

set.seed(8)
seeds <- sample(1:10000, 3)

size <- 1500

points <- as.data.frame(mvtnorm::rmvnorm(
  n = size,
  sigma = diag(size * 4 / (floor(log10(size)) + 1),
    nrow = 2
  )
)) %>%
  rename(
    x = V1,
    y = V2
  ) %>%
  mutate(id = row_number())

points <- get_vectors(points, seeds) %>%
  mutate(distance = gen_simplex(points$x,
                                points$y,
                                frequency = .01,
                                seed = seeds[3])) %>%
  mutate(angle = (atan2(y_direction, x_direction) * 180/pi) %% 360 + hue_turn,
         percentage = pnorm(distance, mean = 0, sd = sd(distance)) * 100) %>%
  rowwise() %>%
  mutate(hex_color = flowfieldfigments::get_color(angle, percentage)) %>%
  ungroup() %>%
  mutate(x_color = percentage * cos(angle * pi/180),
         y_color = percentage * sin(angle * pi/180),
         time = 0) %>%
  filter(!is.na(hex_color))

ggplot() +
  geom_polygon(data = circle,
               aes(x, y),
               color = "black",
               fill = "NA") +
  geom_point(data = points,
             aes(x_color, y_color, color = hex_color)) +
  scale_color_identity() +
  coord_equal() +
  theme_void()

::: ::: {}

set.seed(9)
seeds <- sample(1:10000, 3)

size <- 1500

points <- as.data.frame(mvtnorm::rmvnorm(
  n = size,
  sigma = diag(size * 4 / (floor(log10(size)) + 1),
    nrow = 2
  )
)) %>%
  rename(
    x = V1,
    y = V2
  ) %>%
  mutate(id = row_number())

points <- get_vectors(points, seeds) %>%
  mutate(distance = gen_simplex(points$x,
                                points$y,
                                frequency = .01,
                                seed = seeds[3])) %>%
  mutate(angle = (atan2(y_direction, x_direction) * 180/pi) %% 360 + hue_turn,
         percentage = pnorm(distance, mean = 0, sd = sd(distance)) * 100) %>%
  rowwise() %>%
  mutate(hex_color = flowfieldfigments::get_color(angle, percentage)) %>%
  ungroup() %>%
  mutate(x_color = percentage * cos(angle * pi/180),
         y_color = percentage * sin(angle * pi/180),
         time = 0) %>%
  filter(!is.na(hex_color))

ggplot() +
  geom_polygon(data = circle,
               aes(x, y),
               color = "black",
               fill = "NA") +
  geom_point(data = points,
             aes(x_color, y_color, color = hex_color)) +
  scale_color_identity() +
  coord_equal() +
  theme_void()

::: ::: {}

set.seed(10)
seeds <- sample(1:10000, 3)

size <- 1500

points <- as.data.frame(mvtnorm::rmvnorm(
  n = size,
  sigma = diag(size * 4 / (floor(log10(size)) + 1),
    nrow = 2
  )
)) %>%
  rename(
    x = V1,
    y = V2
  ) %>%
  mutate(id = row_number())

points <- get_vectors(points, seeds) %>%
  mutate(distance = gen_simplex(points$x,
                                points$y,
                                frequency = .01,
                                seed = seeds[3])) %>%
  mutate(angle = (atan2(y_direction, x_direction) * 180/pi) %% 360 + hue_turn,
         percentage = pnorm(distance, mean = 0, sd = sd(distance)) * 100) %>%
  rowwise() %>%
  mutate(hex_color = flowfieldfigments::get_color(angle, percentage)) %>%
  ungroup() %>%
  mutate(x_color = percentage * cos(angle * pi/180),
         y_color = percentage * sin(angle * pi/180),
         time = 0) %>%
  filter(!is.na(hex_color))

ggplot() +
  geom_polygon(data = circle,
               aes(x, y),
               color = "black",
               fill = "NA") +
  geom_point(data = points,
             aes(x_color, y_color, color = hex_color)) +
  scale_color_identity() +
  coord_equal() +
  theme_void()

::: ::::

Spiral

:::: {style="display: grid; grid-template-columns: 1fr 1fr 1fr; align-items: center; justify-items: center; grid-column-gap: 1px;"} ::: {}

set.seed(11)
seeds <- sample(1:10000, 3)

size <- 50

points <- data.frame(
  x = sqrt(seq(1, size)) * cos(golden * seq(1, size)) * 2.5,
  y = sqrt(seq(1, size)) * sin(golden * seq(1, size)) * 2.5
) %>%
  mutate(id = row_number())

points <- get_vectors(points, seeds) %>%
  mutate(distance = gen_simplex(points$x,
                                points$y,
                                frequency = .01,
                                seed = seeds[3])) %>%
  mutate(angle = (atan2(y_direction, x_direction) * 180/pi) %% 360 + hue_turn,
         percentage = pnorm(distance, mean = 0, sd = sd(distance)) * 100) %>%
  rowwise() %>%
  mutate(hex_color = flowfieldfigments::get_color(angle, percentage)) %>%
  ungroup() %>%
  mutate(x_color = percentage * cos(angle * pi/180),
         y_color = percentage * sin(angle * pi/180),
         time = 0) %>%
  filter(!is.na(hex_color))

ggplot() +
  geom_polygon(data = circle,
               aes(x, y),
               color = "black",
               fill = "NA") +
  geom_point(data = points,
             aes(x_color, y_color, color = hex_color)) +
  scale_color_identity() +
  coord_equal() +
  theme_void()

::: ::: {}

set.seed(12)
seeds <- sample(1:10000, 3)

size <- 50

points <- data.frame(
  x = sqrt(seq(1, size)) * cos(golden * seq(1, size)) * 2.5,
  y = sqrt(seq(1, size)) * sin(golden * seq(1, size)) * 2.5
) %>%
  mutate(id = row_number())

points <- get_vectors(points, seeds) %>%
  mutate(distance = gen_simplex(points$x,
                                points$y,
                                frequency = .01,
                                seed = seeds[3])) %>%
  mutate(angle = (atan2(y_direction, x_direction) * 180/pi) %% 360 + hue_turn,
         percentage = pnorm(distance, mean = 0, sd = sd(distance)) * 100) %>%
  rowwise() %>%
  mutate(hex_color = flowfieldfigments::get_color(angle, percentage)) %>%
  ungroup() %>%
  mutate(x_color = percentage * cos(angle * pi/180),
         y_color = percentage * sin(angle * pi/180),
         time = 0) %>%
  filter(!is.na(hex_color))

ggplot() +
  geom_polygon(data = circle,
               aes(x, y),
               color = "black",
               fill = "NA") +
  geom_point(data = points,
             aes(x_color, y_color, color = hex_color)) +
  scale_color_identity() +
  coord_equal() +
  theme_void()

::: ::: {}

set.seed(13)
seeds <- sample(1:10000, 3)

size <- 50

points <- data.frame(
  x = sqrt(seq(1, size)) * cos(golden * seq(1, size)) * 2.5,
  y = sqrt(seq(1, size)) * sin(golden * seq(1, size)) * 2.5
) %>%
  mutate(id = row_number())

points <- get_vectors(points, seeds) %>%
  mutate(distance = gen_simplex(points$x,
                                points$y,
                                frequency = .01,
                                seed = seeds[3])) %>%
  mutate(angle = (atan2(y_direction, x_direction) * 180/pi) %% 360 + hue_turn,
         percentage = pnorm(distance, mean = 0, sd = sd(distance)) * 100) %>%
  rowwise() %>%
  mutate(hex_color = flowfieldfigments::get_color(angle, percentage)) %>%
  ungroup() %>%
  mutate(x_color = percentage * cos(angle * pi/180),
         y_color = percentage * sin(angle * pi/180),
         time = 0) %>%
  filter(!is.na(hex_color))

ggplot() +
  geom_polygon(data = circle,
               aes(x, y),
               color = "black",
               fill = "NA") +
  geom_point(data = points,
             aes(x_color, y_color, color = hex_color)) +
  scale_color_identity() +
  coord_equal() +
  theme_void()

::: ::: {}

set.seed(14)
seeds <- sample(1:10000, 3)

size <- 750

points <- data.frame(
  x = sqrt(seq(1, size)) * cos(golden * seq(1, size)) * 2.5,
  y = sqrt(seq(1, size)) * sin(golden * seq(1, size)) * 2.5
) %>%
  mutate(id = row_number())

points <- get_vectors(points, seeds) %>%
  mutate(distance = gen_simplex(points$x,
                                points$y,
                                frequency = .01,
                                seed = seeds[3])) %>%
  mutate(angle = (atan2(y_direction, x_direction) * 180/pi) %% 360 + hue_turn,
         percentage = pnorm(distance, mean = 0, sd = sd(distance)) * 100) %>%
  rowwise() %>%
  mutate(hex_color = flowfieldfigments::get_color(angle, percentage)) %>%
  ungroup() %>%
  mutate(x_color = percentage * cos(angle * pi/180),
         y_color = percentage * sin(angle * pi/180),
         time = 0) %>%
  filter(!is.na(hex_color))

ggplot() +
  geom_polygon(data = circle,
               aes(x, y),
               color = "black",
               fill = "NA") +
  geom_point(data = points,
             aes(x_color, y_color, color = hex_color)) +
  scale_color_identity() +
  coord_equal() +
  theme_void()

::: ::: {}

set.seed(15)
seeds <- sample(1:10000, 3)

size <- 750

points <- data.frame(
  x = sqrt(seq(1, size)) * cos(golden * seq(1, size)) * 2.5,
  y = sqrt(seq(1, size)) * sin(golden * seq(1, size)) * 2.5
) %>%
  mutate(id = row_number())

points <- get_vectors(points, seeds) %>%
  mutate(distance = gen_simplex(points$x,
                                points$y,
                                frequency = .01,
                                seed = seeds[3])) %>%
  mutate(angle = (atan2(y_direction, x_direction) * 180/pi) %% 360 + hue_turn,
         percentage = pnorm(distance, mean = 0, sd = sd(distance)) * 100) %>%
  rowwise() %>%
  mutate(hex_color = flowfieldfigments::get_color(angle, percentage)) %>%
  ungroup() %>%
  mutate(x_color = percentage * cos(angle * pi/180),
         y_color = percentage * sin(angle * pi/180),
         time = 0) %>%
  filter(!is.na(hex_color))

ggplot() +
  geom_polygon(data = circle,
               aes(x, y),
               color = "black",
               fill = "NA") +
  geom_point(data = points,
             aes(x_color, y_color, color = hex_color)) +
  scale_color_identity() +
  coord_equal() +
  theme_void()

::: ::: {}

set.seed(16)
seeds <- sample(1:10000, 3)

size <- 750

points <- data.frame(
  x = sqrt(seq(1, size)) * cos(golden * seq(1, size)) * 2.5,
  y = sqrt(seq(1, size)) * sin(golden * seq(1, size)) * 2.5
) %>%
  mutate(id = row_number())

points <- get_vectors(points, seeds) %>%
  mutate(distance = gen_simplex(points$x,
                                points$y,
                                frequency = .01,
                                seed = seeds[3])) %>%
  mutate(angle = (atan2(y_direction, x_direction) * 180/pi) %% 360 + hue_turn,
         percentage = pnorm(distance, mean = 0, sd = sd(distance)) * 100) %>%
  rowwise() %>%
  mutate(hex_color = flowfieldfigments::get_color(angle, percentage)) %>%
  ungroup() %>%
  mutate(x_color = percentage * cos(angle * pi/180),
         y_color = percentage * sin(angle * pi/180),
         time = 0) %>%
  filter(!is.na(hex_color))

ggplot() +
  geom_polygon(data = circle,
               aes(x, y),
               color = "black",
               fill = "NA") +
  geom_point(data = points,
             aes(x_color, y_color, color = hex_color)) +
  scale_color_identity() +
  coord_equal() +
  theme_void()

::: ::: {}

set.seed(17)
seeds <- sample(1:10000, 3)

size <- 1500

points <- data.frame(
  x = sqrt(seq(1, size)) * cos(golden * seq(1, size)) * 2.5,
  y = sqrt(seq(1, size)) * sin(golden * seq(1, size)) * 2.5
) %>%
  mutate(id = row_number())

points <- get_vectors(points, seeds) %>%
  mutate(distance = gen_simplex(points$x,
                                points$y,
                                frequency = .01,
                                seed = seeds[3])) %>%
  mutate(angle = (atan2(y_direction, x_direction) * 180/pi) %% 360 + hue_turn,
         percentage = pnorm(distance, mean = 0, sd = sd(distance)) * 100) %>%
  rowwise() %>%
  mutate(hex_color = flowfieldfigments::get_color(angle, percentage)) %>%
  ungroup() %>%
  mutate(x_color = percentage * cos(angle * pi/180),
         y_color = percentage * sin(angle * pi/180),
         time = 0) %>%
  filter(!is.na(hex_color))

ggplot() +
  geom_polygon(data = circle,
               aes(x, y),
               color = "black",
               fill = "NA") +
  geom_point(data = points,
             aes(x_color, y_color, color = hex_color)) +
  scale_color_identity() +
  coord_equal() +
  theme_void()

::: ::: {}

set.seed(18)
seeds <- sample(1:10000, 3)

size <- 1500

points <- data.frame(
  x = sqrt(seq(1, size)) * cos(golden * seq(1, size)) * 2.5,
  y = sqrt(seq(1, size)) * sin(golden * seq(1, size)) * 2.5
) %>%
  mutate(id = row_number())

points <- get_vectors(points, seeds) %>%
  mutate(distance = gen_simplex(points$x,
                                points$y,
                                frequency = .01,
                                seed = seeds[3])) %>%
  mutate(angle = (atan2(y_direction, x_direction) * 180/pi) %% 360 + hue_turn,
         percentage = pnorm(distance, mean = 0, sd = sd(distance)) * 100) %>%
  rowwise() %>%
  mutate(hex_color = flowfieldfigments::get_color(angle, percentage)) %>%
  ungroup() %>%
  mutate(x_color = percentage * cos(angle * pi/180),
         y_color = percentage * sin(angle * pi/180),
         time = 0) %>%
  filter(!is.na(hex_color))

ggplot() +
  geom_polygon(data = circle,
               aes(x, y),
               color = "black",
               fill = "NA") +
  geom_point(data = points,
             aes(x_color, y_color, color = hex_color)) +
  scale_color_identity() +
  coord_equal() +
  theme_void()

::: ::: {}

set.seed(19)
seeds <- sample(1:10000, 3)

size <- 1500

points <- data.frame(
  x = sqrt(seq(1, size)) * cos(golden * seq(1, size)) * 2.5,
  y = sqrt(seq(1, size)) * sin(golden * seq(1, size)) * 2.5
) %>%
  mutate(id = row_number())

points <- get_vectors(points, seeds) %>%
  mutate(distance = gen_simplex(points$x,
                                points$y,
                                frequency = .01,
                                seed = seeds[3])) %>%
  mutate(angle = (atan2(y_direction, x_direction) * 180/pi) %% 360 + hue_turn,
         percentage = pnorm(distance, mean = 0, sd = sd(distance)) * 100) %>%
  rowwise() %>%
  mutate(hex_color = flowfieldfigments::get_color(angle, percentage)) %>%
  ungroup() %>%
  mutate(x_color = percentage * cos(angle * pi/180),
         y_color = percentage * sin(angle * pi/180),
         time = 0) %>%
  filter(!is.na(hex_color))

ggplot() +
  geom_polygon(data = circle,
               aes(x, y),
               color = "black",
               fill = "NA") +
  geom_point(data = points,
             aes(x_color, y_color, color = hex_color)) +
  scale_color_identity() +
  coord_equal() +
  theme_void()

::: ::::

Diamond

:::: {style="display: grid; grid-template-columns: 1fr 1fr 1fr; align-items: center; justify-items: center; grid-column-gap: 1px;"} ::: {}

set.seed(20)
seeds <- sample(1:10000, 3)

size <- 50

grid_width <- ifelse(size <= 1500/2, ceiling(sqrt(size)), floor(sqrt(size)))

points <- expand_grid(x_start = seq(1, grid_width) - (grid_width / 2) - .5,
                      y_start = seq(1, grid_width) - (grid_width / 2) - .5) %>%
  mutate(x = (x_start * cos(45 * pi/180) - y_start * sin(45 * pi/180)) * 5,
         y = (x_start * sin(45 * pi/180) + y_start * cos(45 * pi/180)) * 5,
         id = row_number()) %>%
  select(-x_start, -y_start)

points <- get_vectors(points, seeds) %>%
  mutate(distance = gen_simplex(points$x,
                                points$y,
                                frequency = .01,
                                seed = seeds[3])) %>%
  mutate(angle = (atan2(y_direction, x_direction) * 180/pi) %% 360 + hue_turn,
         percentage = pnorm(distance, mean = 0, sd = sd(distance)) * 100) %>%
  rowwise() %>%
  mutate(hex_color = flowfieldfigments::get_color(angle, percentage)) %>%
  ungroup() %>%
  mutate(x_color = percentage * cos(angle * pi/180),
         y_color = percentage * sin(angle * pi/180),
         time = 0) %>%
  filter(!is.na(hex_color))

ggplot() +
  geom_polygon(data = circle,
               aes(x, y),
               color = "black",
               fill = "NA") +
  geom_point(data = points,
             aes(x_color, y_color, color = hex_color)) +
  scale_color_identity() +
  coord_equal() +
  theme_void()

::: ::: {}

set.seed(21)
seeds <- sample(1:10000, 3)

size <- 50

grid_width <- ifelse(size <= 1500/2, ceiling(sqrt(size)), floor(sqrt(size)))

points <- expand_grid(x_start = seq(1, grid_width) - (grid_width / 2) - .5,
                      y_start = seq(1, grid_width) - (grid_width / 2) - .5) %>%
  mutate(x = (x_start * cos(45 * pi/180) - y_start * sin(45 * pi/180)) * 5,
         y = (x_start * sin(45 * pi/180) + y_start * cos(45 * pi/180)) * 5,
         id = row_number()) %>%
  select(-x_start, -y_start)

points <- get_vectors(points, seeds) %>%
  mutate(distance = gen_simplex(points$x,
                                points$y,
                                frequency = .01,
                                seed = seeds[3])) %>%
  mutate(angle = (atan2(y_direction, x_direction) * 180/pi) %% 360 + hue_turn,
         percentage = pnorm(distance, mean = 0, sd = sd(distance)) * 100) %>%
  rowwise() %>%
  mutate(hex_color = flowfieldfigments::get_color(angle, percentage)) %>%
  ungroup() %>%
  mutate(x_color = percentage * cos(angle * pi/180),
         y_color = percentage * sin(angle * pi/180),
         time = 0) %>%
  filter(!is.na(hex_color))

ggplot() +
  geom_polygon(data = circle,
               aes(x, y),
               color = "black",
               fill = "NA") +
  geom_point(data = points,
             aes(x_color, y_color, color = hex_color)) +
  scale_color_identity() +
  coord_equal() +
  theme_void()

::: ::: {}

set.seed(22)
seeds <- sample(1:10000, 3)

size <- 50

grid_width <- ifelse(size <= 1500/2, ceiling(sqrt(size)), floor(sqrt(size)))

points <- expand_grid(x_start = seq(1, grid_width) - (grid_width / 2) - .5,
                      y_start = seq(1, grid_width) - (grid_width / 2) - .5) %>%
  mutate(x = (x_start * cos(45 * pi/180) - y_start * sin(45 * pi/180)) * 5,
         y = (x_start * sin(45 * pi/180) + y_start * cos(45 * pi/180)) * 5,
         id = row_number()) %>%
  select(-x_start, -y_start)

points <- get_vectors(points, seeds) %>%
  mutate(distance = gen_simplex(points$x,
                                points$y,
                                frequency = .01,
                                seed = seeds[3])) %>%
  mutate(angle = (atan2(y_direction, x_direction) * 180/pi) %% 360 + hue_turn,
         percentage = pnorm(distance, mean = 0, sd = sd(distance)) * 100) %>%
  rowwise() %>%
  mutate(hex_color = flowfieldfigments::get_color(angle, percentage)) %>%
  ungroup() %>%
  mutate(x_color = percentage * cos(angle * pi/180),
         y_color = percentage * sin(angle * pi/180),
         time = 0) %>%
  filter(!is.na(hex_color))

ggplot() +
  geom_polygon(data = circle,
               aes(x, y),
               color = "black",
               fill = "NA") +
  geom_point(data = points,
             aes(x_color, y_color, color = hex_color)) +
  scale_color_identity() +
  coord_equal() +
  theme_void()

::: ::: {}

set.seed(23)
seeds <- sample(1:10000, 3)

size <- 750

grid_width <- ifelse(size <= 1500/2, ceiling(sqrt(size)), floor(sqrt(size)))

points <- expand_grid(x_start = seq(1, grid_width) - (grid_width / 2) - .5,
                      y_start = seq(1, grid_width) - (grid_width / 2) - .5) %>%
  mutate(x = (x_start * cos(45 * pi/180) - y_start * sin(45 * pi/180)) * 5,
         y = (x_start * sin(45 * pi/180) + y_start * cos(45 * pi/180)) * 5,
         id = row_number()) %>%
  select(-x_start, -y_start)

points <- get_vectors(points, seeds) %>%
  mutate(distance = gen_simplex(points$x,
                                points$y,
                                frequency = .01,
                                seed = seeds[3])) %>%
  mutate(angle = (atan2(y_direction, x_direction) * 180/pi) %% 360 + hue_turn,
         percentage = pnorm(distance, mean = 0, sd = sd(distance)) * 100) %>%
  rowwise() %>%
  mutate(hex_color = flowfieldfigments::get_color(angle, percentage)) %>%
  ungroup() %>%
  mutate(x_color = percentage * cos(angle * pi/180),
         y_color = percentage * sin(angle * pi/180),
         time = 0) %>%
  filter(!is.na(hex_color))

ggplot() +
  geom_polygon(data = circle,
               aes(x, y),
               color = "black",
               fill = "NA") +
  geom_point(data = points,
             aes(x_color, y_color, color = hex_color)) +
  scale_color_identity() +
  coord_equal() +
  theme_void()

::: ::: {}

set.seed(24)
seeds <- sample(1:10000, 3)

size <- 750

grid_width <- ifelse(size <= 1500/2, ceiling(sqrt(size)), floor(sqrt(size)))

points <- expand_grid(x_start = seq(1, grid_width) - (grid_width / 2) - .5,
                      y_start = seq(1, grid_width) - (grid_width / 2) - .5) %>%
  mutate(x = (x_start * cos(45 * pi/180) - y_start * sin(45 * pi/180)) * 5,
         y = (x_start * sin(45 * pi/180) + y_start * cos(45 * pi/180)) * 5,
         id = row_number()) %>%
  select(-x_start, -y_start)

points <- get_vectors(points, seeds) %>%
  mutate(distance = gen_simplex(points$x,
                                points$y,
                                frequency = .01,
                                seed = seeds[3])) %>%
  mutate(angle = (atan2(y_direction, x_direction) * 180/pi) %% 360 + hue_turn,
         percentage = pnorm(distance, mean = 0, sd = sd(distance)) * 100) %>%
  rowwise() %>%
  mutate(hex_color = flowfieldfigments::get_color(angle, percentage)) %>%
  ungroup() %>%
  mutate(x_color = percentage * cos(angle * pi/180),
         y_color = percentage * sin(angle * pi/180),
         time = 0) %>%
  filter(!is.na(hex_color))

ggplot() +
  geom_polygon(data = circle,
               aes(x, y),
               color = "black",
               fill = "NA") +
  geom_point(data = points,
             aes(x_color, y_color, color = hex_color)) +
  scale_color_identity() +
  coord_equal() +
  theme_void()

::: ::: {}

set.seed(25)
seeds <- sample(1:10000, 3)

size <- 750

grid_width <- ifelse(size <= 1500/2, ceiling(sqrt(size)), floor(sqrt(size)))

points <- expand_grid(x_start = seq(1, grid_width) - (grid_width / 2) - .5,
                      y_start = seq(1, grid_width) - (grid_width / 2) - .5) %>%
  mutate(x = (x_start * cos(45 * pi/180) - y_start * sin(45 * pi/180)) * 5,
         y = (x_start * sin(45 * pi/180) + y_start * cos(45 * pi/180)) * 5,
         id = row_number()) %>%
  select(-x_start, -y_start)

points <- get_vectors(points, seeds) %>%
  mutate(distance = gen_simplex(points$x,
                                points$y,
                                frequency = .01,
                                seed = seeds[3])) %>%
  mutate(angle = (atan2(y_direction, x_direction) * 180/pi) %% 360 + hue_turn,
         percentage = pnorm(distance, mean = 0, sd = sd(distance)) * 100) %>%
  rowwise() %>%
  mutate(hex_color = flowfieldfigments::get_color(angle, percentage)) %>%
  ungroup() %>%
  mutate(x_color = percentage * cos(angle * pi/180),
         y_color = percentage * sin(angle * pi/180),
         time = 0) %>%
  filter(!is.na(hex_color))

ggplot() +
  geom_polygon(data = circle,
               aes(x, y),
               color = "black",
               fill = "NA") +
  geom_point(data = points,
             aes(x_color, y_color, color = hex_color)) +
  scale_color_identity() +
  coord_equal() +
  theme_void()

::: ::: {}

set.seed(26)
seeds <- sample(1:10000, 3)

size <- 1500

grid_width <- ifelse(size <= 1500/2, ceiling(sqrt(size)), floor(sqrt(size)))

points <- expand_grid(x_start = seq(1, grid_width) - (grid_width / 2) - .5,
                      y_start = seq(1, grid_width) - (grid_width / 2) - .5) %>%
  mutate(x = (x_start * cos(45 * pi/180) - y_start * sin(45 * pi/180)) * 5,
         y = (x_start * sin(45 * pi/180) + y_start * cos(45 * pi/180)) * 5,
         id = row_number()) %>%
  select(-x_start, -y_start)

points <- get_vectors(points, seeds) %>%
  mutate(distance = gen_simplex(points$x,
                                points$y,
                                frequency = .01,
                                seed = seeds[3])) %>%
  mutate(angle = (atan2(y_direction, x_direction) * 180/pi) %% 360 + hue_turn,
         percentage = pnorm(distance, mean = 0, sd = sd(distance)) * 100) %>%
  rowwise() %>%
  mutate(hex_color = flowfieldfigments::get_color(angle, percentage)) %>%
  ungroup() %>%
  mutate(x_color = percentage * cos(angle * pi/180),
         y_color = percentage * sin(angle * pi/180),
         time = 0) %>%
  filter(!is.na(hex_color))

ggplot() +
  geom_polygon(data = circle,
               aes(x, y),
               color = "black",
               fill = "NA") +
  geom_point(data = points,
             aes(x_color, y_color, color = hex_color)) +
  scale_color_identity() +
  coord_equal() +
  theme_void()

::: ::: {}

set.seed(27)
seeds <- sample(1:10000, 3)

size <- 1500

grid_width <- ifelse(size <= 1500/2, ceiling(sqrt(size)), floor(sqrt(size)))

points <- expand_grid(x_start = seq(1, grid_width) - (grid_width / 2) - .5,
                      y_start = seq(1, grid_width) - (grid_width / 2) - .5) %>%
  mutate(x = (x_start * cos(45 * pi/180) - y_start * sin(45 * pi/180)) * 5,
         y = (x_start * sin(45 * pi/180) + y_start * cos(45 * pi/180)) * 5,
         id = row_number()) %>%
  select(-x_start, -y_start)

points <- get_vectors(points, seeds) %>%
  mutate(distance = gen_simplex(points$x,
                                points$y,
                                frequency = .01,
                                seed = seeds[3])) %>%
  mutate(angle = (atan2(y_direction, x_direction) * 180/pi) %% 360 + hue_turn,
         percentage = pnorm(distance, mean = 0, sd = sd(distance)) * 100) %>%
  rowwise() %>%
  mutate(hex_color = flowfieldfigments::get_color(angle, percentage)) %>%
  ungroup() %>%
  mutate(x_color = percentage * cos(angle * pi/180),
         y_color = percentage * sin(angle * pi/180),
         time = 0) %>%
  filter(!is.na(hex_color))

ggplot() +
  geom_polygon(data = circle,
               aes(x, y),
               color = "black",
               fill = "NA") +
  geom_point(data = points,
             aes(x_color, y_color, color = hex_color)) +
  scale_color_identity() +
  coord_equal() +
  theme_void()

::: ::: {}

set.seed(28)
seeds <- sample(1:10000, 3)

size <- 1500

grid_width <- ifelse(size <= 1500/2, ceiling(sqrt(size)), floor(sqrt(size)))

points <- expand_grid(x_start = seq(1, grid_width) - (grid_width / 2) - .5,
                      y_start = seq(1, grid_width) - (grid_width / 2) - .5) %>%
  mutate(x = (x_start * cos(45 * pi/180) - y_start * sin(45 * pi/180)) * 5,
         y = (x_start * sin(45 * pi/180) + y_start * cos(45 * pi/180)) * 5,
         id = row_number()) %>%
  select(-x_start, -y_start)

points <- get_vectors(points, seeds) %>%
  mutate(distance = gen_simplex(points$x,
                                points$y,
                                frequency = .01,
                                seed = seeds[3])) %>%
  mutate(angle = (atan2(y_direction, x_direction) * 180/pi) %% 360 + hue_turn,
         percentage = pnorm(distance, mean = 0, sd = sd(distance)) * 100) %>%
  rowwise() %>%
  mutate(hex_color = flowfieldfigments::get_color(angle, percentage)) %>%
  ungroup() %>%
  mutate(x_color = percentage * cos(angle * pi/180),
         y_color = percentage * sin(angle * pi/180),
         time = 0) %>%
  filter(!is.na(hex_color))

ggplot() +
  geom_polygon(data = circle,
               aes(x, y),
               color = "black",
               fill = "NA") +
  geom_point(data = points,
             aes(x_color, y_color, color = hex_color)) +
  scale_color_identity() +
  coord_equal() +
  theme_void()