# Mathematical aRt

Marcus Volz is a research fellow at the University of Melbourne, studying geometric networks, optimisation and computational geometry. He’s interested in visualisation, and always looking for opportunities to represent complex information in novel ways to accelerate learning and uncover the unexpected.

One of Marcus’ hobbies is the visualization of mathematical patterns and statistical algorithms via R. He has a whole portfolio full of them, including a Github page with all the associated R code. For my recent promotion, my girlfriend asked Marcus to generate a K-nearest neighbors visual and she had it printed on a large canvas. The picture contains about 10.000 points, randomly uniformly distributed across x and y, connected by lines with their closest other points. Marcus shared the code to generate such k-nearest neighbor algorithm plots here on Github. So if you know your way around R, you could make your own version:

```#' k-nearest neighbour graph
#'
#' Computes a k-nearest neighbour graph for a given set of points. Refer to the \href{https://en.wikipedia.org/wiki/Nearest_neighbor_graph}{Wikipedia article} for details.
#' @param points A data frame with x, y coordinates for the points
#' @param k Number of neighbours
#' @keywords nearest neightbour graph
#' @export
#' @examples
#' k_nearest_neighbour_graph()

k_nearest_neighbour_graph <- function(points, k=8) {
get_k_nearest <- function(points, ptnum, k) {
xi <- points\$x[ptnum]
yi <- points\$y[ptnum]     points %>%
dplyr::mutate(dist = sqrt((x - xi)^2 + (y - yi)^2)) %>%
dplyr::arrange(dist) %>%
dplyr::filter(row_number() %in% seq(2, k+1)) %>%
dplyr::mutate(xend = xi, yend = yi)
}

1:nrow(points) %>%
purrr::map_df(~get_k_nearest(points, ., k))
}```

Those less versed in R can use Marcus package `mathart`. With this package, Marcus shares many more visual depictions of cool algorithms! You can install the package and several dependencies with the following lines of code:

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

Subsequently, you can visualize all kinds of cool stuff, like for instance rapidly exploring random trees (see this Wikipedia article for details):

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

# Create plot
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)
```

This k-d tree (see this Wikipedia article for details) is also amazing:

```result <- kdtree(mathart::points)

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

This page of Marcus’ mathart Github repository contains the code exact code for these and many other visualizations of algorithms and statistical phenomena. Do check it out if you’re interested!

Also, check out the “Fun” section of my R tips and tricks list for more cool visuals you can generate in R!

## One thought on “Mathematical aRt”

1. Timothy Schmidt says:

Beautiful! Thank you! 🙂

Like