La luna es un pozo chico

las flores no valen nada

lo que valen son tus brazos

cuando de noche me abrazan

(Zorongo Gitano, Carmen Linares)

When I publish a post showing my drawings, I use to place some outputs, give some highlights about the techniques involved as well as a link to the R code that I write to generate them. That’s my typical *generative-art* post (here you have an example of it). I think that my audience knows to program in R and is curious enough to run and modify the code by themselves to generate their own outputs. Today I will try to be more *educational* and will explain step by step how you can obtain drawings like these:

There are two reasons for this decision:

- It can illustrate quite well my
*mental journey*from a simple idea to what I think is a interesting enough experiment to publish. - I think that this experiment is a good example of the use of
`accumulate`

, a very useful function from the*life-changing*`purrr`

package.

Here we go: there are many ways of drawing a pentagon in R. Following you will find a piece of code that does it using `accumulate`

function from `purrr`

package. I will use only two libraries for this experiment: `ggplot2`

and `purrr`

so I will just load in the `tidyverse`

(both libraries take part of it):

```
library(tidyverse)
pentagon <- tibble(
x = accumulate(1:4, ~.x+cos(.y*2*pi/5), .init = 0),
y = accumulate(1:4, ~.x+sin(.y*2*pi/5), .init = 0),
xend = accumulate(2:5, ~.x+cos(.y*2*pi/5), .init = cos(2*pi/5)),
yend = accumulate(2:5, ~.x+sin(.y*2*pi/5), .init = sin(2*pi/5)))
ggplot(pentagon)+
geom_segment(aes(x=x, y=y, xend=xend, yend=yend))+
coord_equal()+
theme_void()
```

The function `accumulate`

applies sequentially some function a number of times storing all the intermediate results. When I say sequentially I mean that the input of any step is the output of the prevoius one. The `accumulate`

function uses internally two important arguments called `.x`

and `.y`

: my own way to understand its significance is that `.x`

is the previous value of the output vector and `.y`

is the previous value of the one which controls the iteration. Let’s see a example: imagine that I want to create a vector with the first 10 natural numbers. This is an option:

```
> accumulate(1:10, ~.y)
[1] 1 2 3 4 5 6 7 8 9 10
```

The vector which controls the iteration in this case is `1:10`

and `.y`

are the values of it so I just have to define a function wich returns that values and this is as simple as `~.y`

. The first iteration takes the first element of that vector. This is another way to do it:

```
> accumulate(1:10, ~.x+1)
[1] 1 2 3 4 5 6 7 8 9 10
```

To replicate the same output with `.x`

I have to change a bit the function to `~.x+1`

because if not, it will always return `1`

. Remember that `.x`

is the previous output of the function and it is initialized with `1`

(the first value of the vector `1:10`

). Intead of initializing `.x`

with the first value of the vector of the first argument of `accumulate`

, you can define exactly its first value using `.init`

:

```
accumulate(2:10, ~.y, .init = 1)
accumulate(1:9, ~.x+1, .init = 1)
```

Note that using `.init`

I have to change the vector to reproduce the same output as before. I hope now you will understand how I generated the initial and ending points of the previous pentagon. Some points to help you if not:

- I generate a tibble with
`5`

rows, each of one defines a different segment of the pentagon - First segments starts at
`(0,0)`

- The rotating angle is equal to
`2*pi/5`

- The ending point of each segment becomes the starting point of the following one

The next step is to *encapsulate* this into a function to draw regular polygons with any given number of edges. I only have to generalize the number of steps and the rotating angle of `accumulate`

:

```
polygon <- function(n) {
tibble(
x = accumulate(1:(n-1), ~.x+cos(.y*2*pi/n), .init = 0),
y = accumulate(1:(n-1), ~.x+sin(.y*2*pi/n), .init = 0),
xend = accumulate(2:n, ~.x+cos(.y*2*pi/n), .init = cos(2*pi/n)),
yend = accumulate(2:n, ~.x+sin(.y*2*pi/n), .init = sin(2*pi/n)))
}
ggplot(polygon(6))+
geom_segment(aes(x=x, y=y, xend=xend, yend=yend))+
coord_equal()+
theme_void()
ggplot(polygon(7))+
geom_segment(aes(x=x, y=y, xend=xend, yend=yend))+
coord_equal()+
theme_void()
ggplot(polygon(8))+
geom_segment(aes(x=x, y=y, xend=xend, yend=yend))+
coord_equal()+
theme_void()
ggplot(polygon(9))+
geom_segment(aes(x=x, y=y, xend=xend, yend=yend))+
coord_equal()+
theme_void()
```

Now, let’s place another segment in the middle of each edge, perpendicular to it towards its centre. To do it I *mutate* de data frame to add those segments using *simple* trigonometry: I just have to add `pi/2`

to the angle wich forms the edge, obtained with `atan2`

function:

```
polygon(5) -> df1
df1 %>% mutate(angle = atan2(yend-y, xend-x)+pi/2,
x = 0.5*x+0.5*xend,
y = 0.5*y+0.5*yend,
xend = x+0.2*cos(angle),
yend = y+0.2*sin(angle)) %>%
select(x, y, xend, yend) -> df2
df1 %>% bind_rows(df2) -> df
ggplot(df)+
geom_segment(aes(x=x, y=y, xend=xend, yend=yend))+
coord_equal()+
theme_void()
```

These new segments have longitude equal to `0.2`

, smaller than the original edges of the pentagon. Now, let’s connect the ending points of these perpendicular segments. It is easy using `mutate`

and `first`

functions. Another smaller pentagon appears:

```
polygon(5) -> df1
df1 %>% mutate(angle = atan2(yend-y, xend-x)+pi/2,
x = 0.5*x+0.5*xend,
y = 0.5*y+0.5*yend,
xend = x+0.2*cos(angle),
yend = y+0.2*sin(angle)) %>%
select(x, y, xend, yend) -> df2
df2 %>% mutate(
x=xend,
y=yend,
xend=lead(x, default=first(x)),
yend=lead(y, default=first(y))) %>%
select(x, y, xend, yend) -> df3
df1 %>% bind_rows(df2) %>% bind_rows(df3) -> df
ggplot(df)+
geom_segment(aes(x=x, y=y, xend=xend, yend=yend))+
coord_equal()+
theme_void()
```

Since we are repeating these steps many times, I will write two functions: one to generate perpendicular segments to the edges called `mid_points`

and another one to connect its ending points called `con_points`

. The next code creates both funtions and uses them to add another level to our previous drawing:

```
mid_points <- function(d) {
d %>% mutate(
angle=atan2(yend-y, xend-x) + pi/2,
x=0.5*x+0.5*xend,
y=0.5*y+0.5*yend,
xend=x+0.2*cos(angle),
yend=y+0.2*sin(angle)) %>%
select(x, y, xend, yend)
}
con_points <- function(d) {
d %>% mutate(
x=xend,
y=yend,
xend=lead(x, default=first(x)),
yend=lead(y, default=first(y))) %>%
select(x, y, xend, yend)
}
polygon(5) -> df1
df2 <- mid_points(df1)
df3 <- con_points(df2)
df4 <- mid_points(df3)
df5 <- con_points(df4)
df1 %>%
bind_rows(df2) %>%
bind_rows(df3) %>%
bind_rows(df4) %>%
bind_rows(df5) -> df
ggplot(df)+
geom_segment(aes(x=x, y=y, xend=xend, yend=yend))+
coord_equal()+
theme_void()
```

This pattern is called Sutcliffe pentagon. In the previous step, I did iterations *manually*. The function `accumulate`

can help us to do it automatically. This code reproduces exactly the previous plot:

```
edges <- 5
niter <- 4
polygon(edges) -> df1
accumulate(.f = function(old, y) {
if (y%%2!=0) mid_points(old) else con_points(old)
},
1:niter,
.init=df1) %>%
bind_rows() -> df
ggplot(df)+
geom_segment(aes(x=x, y=y, xend=xend, yend=yend))+
coord_equal()+
theme_void()
```

Substituting `edges`

by `7`

and `niter`

by `6`

as well in the first two rows of the previous code, generates a different pattern, in this case heptagonal:

Let’s start to play with the parameters to change the appearance of the drawings. What if we do not start the perpendicular segments from the midpoints of the edges? It’s easy: we just need to add a parameter that will name `p`

to the function `mid_points`

(`p=0.5`

means starting from the middle). This is our heptagon pattern when `p`

is equal to `0.3`

:

```
mid_points <- function(d, p) {
d %>% mutate(
angle=atan2(yend-y, xend-x) + pi/2,
x=p*x+(1-p)*xend,
y=p*y+(1-p)*yend,
xend=x+0.2*cos(angle),
yend=y+0.2*sin(angle)) %>%
select(x, y, xend, yend)
}
edges <- 7
niter <- 6
polygon(edges) -> df1
accumulate(.f = function(old, y) {
if (y%%2==0) mid_points(old, 0.3) else con_points(old)
},
1:niter,
.init=df1) %>%
bind_rows() -> df
ggplot(df)+
geom_segment(aes(x=x, y=y, xend=xend, yend=yend))+
coord_equal()+
theme_void()
```

Another simple modification is to allow any angle between edges and next iteration segments (perpendicular until now ) so let’s add another parameter, called `a`

, to the`mid_points`

function:

```
mid_points <- function(d, p, a) {
d %>% mutate(
angle=atan2(yend-y, xend-x) + a,
x=p*x+(1-p)*xend,
y=p*y+(1-p)*yend,
xend=x+0.2*cos(angle),
yend=y+0.2*sin(angle)) %>%
select(x, y, xend, yend)
}
edges <- 7
niter <- 18
polygon(edges) -> df1
accumulate(.f = function(old, y) {
if (y%%2!=0) mid_points(old, 0.3, pi/5) else con_points(old)
},
1:niter,
.init=df1) %>%
bind_rows() -> df
ggplot(df)+
geom_segment(aes(x=x, y=y, xend=xend, yend=yend))+
coord_equal()+
theme_void()
```

That’s nice! It looks like a shutter. Now it’s time to change the longitude of the segments starting from the edges (those perpendicular in our first drawings). Now all them measure `0.2`

. I will take advantage of the parameter `y`

of `accumulate`

and apply a user defined function to modify that longitude each iteration. This example uses the identity function (`FUN = function(x) x`

) to increase longitude step by step:

```
mid_points <- function(d, p, a, i, FUN = function(x) x) {
d %>% mutate(
angle=atan2(yend-y, xend-x) + a,
radius=FUN(i),
x=p*x+(1-p)*xend,
y=p*y+(1-p)*yend,
xend=x+radius*cos(angle),
yend=y+radius*sin(angle)) %>%
select(x, y, xend, yend)
}
edges <- 7
niter <- 18
polygon(edges) -> df1
accumulate(.f = function(old, y) {
if (y%%2!=0) mid_points(old, 0.3, pi/5, y) else con_points(old)
},
1:niter,
.init=df1) %>%
bind_rows() -> df
ggplot(df)+
geom_segment(aes(x=x, y=y, xend=xend, yend=yend))+
coord_equal()+
theme_void()
```

What if we increase `niter`

from 18 to 250?

```
edges <- 7
niter <- 250
step <- 2
polygon(edges) -> df1
accumulate(.f = function(old, y) {
if (y%%step!=0) mid_points(old, 0.3, pi/5, y) else con_points(old)
},
1:niter,
.init=df1) %>%
bind_rows() -> df
ggplot(df)+
geom_curve(aes(x=x, y=y, xend=xend, yend=yend),
curvature = 0,
color="black",
alpha=0.1)+
coord_equal()+
theme(legend.position = "none",
panel.background = element_rect(fill="white"),
plot.background = element_rect(fill="white"),
axis.ticks = element_blank(),
panel.grid = element_blank(),
axis.title = element_blank(),
axis.text = element_blank())
```

Not bad, but we can do it better. First of all, note that appart of adding transparency with the parameter `alpha`

inside the `ggplot`

function, I changed the geometry of the plot from `geom_segment`

to `geom_curve`

. Setting `curvature = 0`

as I did generates straight lines so the result is the same as `geom_segment`

but it will give us an additional *degree of freedom* to do our plots. I also changed the `theme_void`

by an explicit customization some of the elements of the plot. Concretely, I want to be able to change the background color. This is the definitive code explained:

```
library(tidyverse)
# This function creates the segments of the original polygon
polygon <- function(n) {
tibble(
x = accumulate(1:(n-1), ~.x+cos(.y*2*pi/n), .init = 0),
y = accumulate(1:(n-1), ~.x+sin(.y*2*pi/n), .init = 0),
xend = accumulate(2:n, ~.x+cos(.y*2*pi/n), .init = cos(2*pi/n)),
yend = accumulate(2:n, ~.x+sin(.y*2*pi/n), .init = sin(2*pi/n)))
}
# This function creates segments from some mid-point of the edges
mid_points <- function(d, p, a, i, FUN = ratio_f) {
d %>% mutate(
angle=atan2(yend-y, xend-x) + a,
radius=FUN(i),
x=p*x+(1-p)*xend,
y=p*y+(1-p)*yend,
xend=x+radius*cos(angle),
yend=y+radius*sin(angle)) %>%
select(x, y, xend, yend)
}
# This function connect the ending points of mid-segments
con_points <- function(d) {
d %>% mutate(
x=xend,
y=yend,
xend=lead(x, default=first(x)),
yend=lead(y, default=first(y))) %>%
select(x, y, xend, yend)
}
edges <- 3 # Number of edges of the original polygon
niter <- 250 # Number of iterations
pond <- 0.24 # Weight to calculate the point on the middle of each edge
step <- 13 # No of times to draw mid-segments before connect ending points
alph <- 0.25 # transparency of curves in geom_curve
angle <- 0.6 # angle of mid-segment with the edge
curv <- 0.1 # Curvature of curves
line_color <- "black" # Color of curves in geom_curve
back_color <- "white" # Background of the ggplot
ratio_f <- function(x) {sin(x)} # To calculate the longitude of mid-segments
# Generation on the fly of the dataset
accumulate(.f = function(old, y) {
if (y%%step!=0) mid_points(old, pond, angle, y) else con_points(old)
}, 1:niter,
.init=polygon(edges)) %>% bind_rows() -> df
# Plot
ggplot(df)+
geom_curve(aes(x=x, y=y, xend=xend, yend=yend),
curvature = curv,
color=line_color,
alpha=alph)+
coord_equal()+
theme(legend.position = "none",
panel.background = element_rect(fill=back_color),
plot.background = element_rect(fill=back_color),
axis.ticks = element_blank(),
panel.grid = element_blank(),
axis.title = element_blank(),
axis.text = element_blank())
```

The next table shows the parameters of each of the previous drawings (from left to right and top to bottom):

edges | niter | pond | step | alph | angle | curv | line_color | back_color | ratio_f | |
---|---|---|---|---|---|---|---|---|---|---|

1 | 4 | 200 | 0.92 | 9 | 0.50 | 6.12 | 0.0 | black | white | function (x) { x } |

2 | 5 | 150 | 0.72 | 13 | 0.35 | 2.96 | 0.0 | black | white | function (x) { sqrt(x) } |

3 | 15 | 250 | 0.67 | 9 | 0.15 | 1.27 | 1.0 | black | white | function (x) { sin(x) } |

4 | 9 | 150 | 0.89 | 14 | 0.35 | 3.23 | 0.0 | black | white | function (x) { sin(x) } |

5 | 5 | 150 | 0.27 | 17 | 0.35 | 4.62 | 0.0 | black | white | function (x) { log(x + 1) } |

6 | 14 | 100 | 0.87 | 14 | 0.15 | 0.57 | -2.0 | black | white | function (x) { 1 – cos(x)^2 } |

7 | 7 | 150 | 0.19 | 6 | 0.45 | 3.59 | 0.0 | black | white | function (x) { 1 – cos(x)^2 } |

8 | 4 | 150 | 0.22 | 10 | 0.45 | 4.78 | 0.0 | black | white | function (x) { 1/x } |

9 | 3 | 250 | 0.24 | 13 | 0.25 | 0.60 | 0.1 | black | white | function (x) { sin(x) } |

You can also play with colors. By the way: this document will help you to choose them by their name. Some examples:

I will not *unveil* the parameters of the previous drawings. Maybe it can encourage you to try by yourself and find your own patterns. If you do, I will love to see them. I hope you enjoy this reading. The code is also available here.