Line-replacement fractals, also known as iterated function system (IFS) fractals, can be created by repeatedly applying iteration rules on curves. One famous example is the Koch snowflake. Here's an R programming example for computing properties of the Koch snowflake fractal:
```R
# Load necessary libraries
library(ggplot2)
# Define the initial segment of the Koch snowflake
initial_segment <- data.frame(x = c(0, 1), y = c(0, 0))
# Function to generate the next level of the Koch snowflake
generate_koch_segment <- function(segment) {
x <- segment$x
y <- segment$y
new_x <- c(x[1], (2*x[1] + x[2]) / 3, (x[1] + x[2]) / 2, (x[1] + 2*x[2]) / 3, x[2])
new_y <- c(y[1], (2*y[1] + y[2]) / 3, (y[1] + y[2]) / 2, (y[1] + 2*y[2]) / 3, y[2])
return(data.frame(x = new_x, y = new_y))
}
# Generate the Koch snowflake by applying the rules iteratively
koch_snowflake <- initial_segment
for (i in 1:5) { # Increase the number of iterations for a more detailed snowflake
koch_snowflake <- lapply(1:(length(koch_snowflake) - 1),
function(j) generate_koch_segment(koch_snowflake[[j]]))
koch_snowflake <- do.call(rbind, koch_snowflake)
}
# Plot the Koch snowflake
ggplot(koch_snowflake, aes(x, y)) +
geom_path() +
labs(title = "Koch Snowflake Fractal", x = "", y = "") +
theme_minimal()
```
This code defines the initial segment of the Koch snowflake and a function to generate the next level of segments based on the Koch fractal's rules. It then iteratively applies these rules to create a detailed snowflake.
You can adjust the number of iterations and the parameters of the fractal to create more complex or detailed fractals. The code also uses the `ggplot2` library to create a visualization of the fractal.
Shape-replacement fractals, also known as L-systems (Lindenmayer systems), are created by repeatedly applying iteration rules on shapes or strings. Let's explore an example of generating a fractal tree using R and compute some properties such as the number of segments and the total length of the tree:
```R
# Load necessary libraries
library(ggplot2)
# Define the initial axiom and rules for the L-system
axiom <- "X"
rules <- list(
"X" = "F+[[X]-X]-F[-FX]+X",
"F" = "FF"
)
# Function to generate the L-system string
generate_l_system <- function(axiom, rules, iterations) {
l_system <- axiom
for (i in 1:iterations) {
l_system <- gsub(pattern = names(rules), replacement = rules, x = l_system)
}
return(l_system)
}
# Function to compute the properties of the L-system
compute_l_system_properties <- function(l_system, length_per_iteration) {
num_segments <- sum(l_system == "F")
total_length <- num_segments * length_per_iteration
return(list(
"Number of Segments" = num_segments,
"Total Length" = total_length
))
}
# Generate the L-system string and compute properties
iterations <- 4 # Adjust the number of iterations for a more detailed tree
l_system_string <- generate_l_system(axiom, rules, iterations)
properties <- compute_l_system_properties(l_system_string, length_per_iteration = 10) # Length per iteration
# Print the properties
for (prop in names(properties)) {
cat(prop, ": ", properties[[prop]], "\n")
}
# Create a visualization of the L-system (fractal tree)
x <- 0
y <- 0
angle <- 90
stack <- data.frame(x = numeric(0), y = numeric(0))
segments <- data.frame(x = numeric(0), y = numeric(0))
for (symbol in strsplit(l_system_string, NULL)[[1]]) {
if (symbol == "F") {
x <- x + cos(angle * pi / 180)
y <- y + sin(angle * pi / 180)
segments <- rbind(segments, data.frame(x = x, y = y))
} else if (symbol == "+") {
angle <- angle + 25
} else if (symbol == "-") {
angle <- angle - 25
} else if (symbol == "[") {
stack <- rbind(stack, data.frame(x = x, y = y))
} else if (symbol == "]") {
n <- nrow(stack)
if (n > 0) {
x <- stack[n, "x"]
y <- stack[n, "y"]
stack <- stack[-n, ]
}
}
}
ggplot(segments, aes(x, y)) +
geom_path() +
labs(title = "Fractal Tree (L-System)") +
theme_minimal()
```
In this code, we define the axiom and rules for an L-system representing a fractal tree. The `generate_l_system` function iteratively applies the rules to generate the L-system string. Then, we compute the number of segments and the total length of the tree using the `compute_l_system_properties` function. Finally, we create a visualization of the fractal tree using ggplot2. You can adjust the number of iterations for a more detailed tree and other parameters to create different shapes.
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