### A Pluto.jl notebook ###
# v1.0.4

using Markdown
using InteractiveUtils

# ╔═╡ bff39008-852b-4671-9afa-198b1315d914
using Images, FileIO, Random

# ╔═╡ 95801422-0e12-442b-8e39-c242890502b2
md"# **MATH 2121 - Fall 2026**
Course demonstration written using the `Julia` programming language and shown using the interactive `Pluto` environment."

# ╔═╡ 6dd1f439-540e-44ed-a92c-dac7c2cc323a


# ╔═╡ 9a7fa844-0e94-11eb-0850-31e34452fd9b
md"""There is a bounded region of space in $\mathbb{R}^n$ that we can associate to any $n \times n$ matrix: namely, consider the *$n$-dimensional parallelogram* (called a  *parallelepiped* when $n=3$) whose edges are the columns of your matrix.
"""

# ╔═╡ 51eb2cec-0e95-11eb-0920-79c8815b98da
md"""Our goal today is to uncover a formula for the volume for this region."""

# ╔═╡ 867f0302-0e95-11eb-2633-8be05070104d


# ╔═╡ c8e33a4c-0e95-11eb-2e77-e96a550786e3
md"""In $2$ dimensions, we usually refer to volume as *area*. 

Consider the following $2\times 2$ matrix:
"""

# ╔═╡ 3e8ba024-0e96-11eb-36a5-61db9665d7d0
A = [-4.22 0.98; 2.28 -4.72]

# ╔═╡ 2dc9055e-0e96-11eb-04a8-cf3d9d9683a7
begin
	using LinearAlgebra
	using Plots
	plotly()
	gr()
	
	function matrix_plot(A)
		p = scatter([0], [0], color = :red, legend = false, label = "origin",
			aspect_ratio=:equal,
			title="Parallelogram of a 2-by-2 matrix", mode = "markers")
		scatter!([A[1, 1]], [A[2, 1]], label = "first column", color = :red, mode = "markers")
		scatter!([A[1, 2]], [A[2, 2]], label = "second column", color = :red, mode = "markers")
		scatter!(
			[A[1, 1] + A[1, 2]], [A[2, 1] + A[2, 2]], 
			label = "sum of columns", color = :red, mode = "markers")

		quiver!(quiver = ([A[1, 1]], [A[2, 1]]), [0], [0], color = :blue,)
		quiver!(quiver = ([A[1, 2]], [A[2, 2]]), [0], [0], color = :blue,)
		plot!(Shape(
				[0, A[1, 1], A[1, 1] + A[1, 2], A[1, 2]],
				[0, A[2, 1], A[2, 1] + A[2, 2], A[2, 2]],), opacity=.25, color = :blue)

		return p
	end
	
	matrix_plot(A)
end

# ╔═╡ d7d4030a-0ea0-11eb-050f-e9e4e781a13c
md"Here is a picture of the parallelogram spanned by the columns of $M$:"

# ╔═╡ 87aa9f88-0ea0-11eb-04da-99c18a359c09
md"How can we characterize this parallelogram? 

Well, every point inside the parallelogram has the form $Av$ for some vector 

$v = \left[\begin{array}{c} v_1 \\ v_2\end{array}\right] \in \mathbb{R}^2$ 

with $0\leq v_1 \leq 1$ and $0\leq v_2 \leq 1$. This suggests a general definition:

The *$n$-dimensional parallelogram* of an $n\times n$ matrix $A$ is the set of all points

$\left\{ Av : v= \left[\begin{array}{c} v_1 \\ v_2 \\ \vdots \\ v_n \end{array}\right]\in\mathbb{R}^n\text{ with } 0\leq v_i \leq 1 \text{ for }i=1,2,\dots,n\right\}.$

Define the *volume* of an $n\times n$ matrix to be the volume of this region.
"

# ╔═╡ 32b5167e-0ea1-11eb-3db0-cdc1d26df891
#=md"Next issue: how is volume defined in $\mathbb{R}^n$?

The definition should obey our intuition that the volume of a rectangular region is 

$\text{length} \times \text{width} \times \text{height} \times \dots$

and volume should be *invariant under translation*.

That is, if $L_1,L_2,\dots,L_n$ are positive numbers then the region 

$\mathrm{rect}(L_1,L_2,\dots,L_n) = \{ v \in \mathbb{R}^n : 0\leq v_i \leq L_i\text{ for }i=1,2,\dots,n\}$

should have volume $L_1\cdot L_2\cdots L_n$. Moreover, any translate of this region

$u + \mathrm{rect}(L_1,L_2,\dots,L_n) = \{ u + v \in \mathbb{R}^n : 0\leq v_i \leq L_i\text{ for }i=1,2,\dots,n\}$

where $u \in \mathbb{R}^n$ is some given element should have the same volume.

Finally, if we have two regions in $\mathbb{R}^n$ that don't overlap, then the volume of their union should be the sum of their individual volumes. "=#

#=md"Here is one definition of the volume of a region $S$ in $\mathbb{R}^n$ that satisfies these properties: choose small positive numbers $L_1,L_2,\dots,L_n > 0$ and fill up as much of $S$ as possible with nonoverlapping translates of the $n$-dimensional rectangle $\mathrm{rect}(L_1,L_2,\dots,L_n)$. If you can fit $N$ such regions completely inside $S$ without any overlap, then you say your *estimated* volume is 

$N \cdot L_1 \cdot L_2\cdots L_n.$

The true *volume* of $S$ is then the *limit superior* of all possible estimated volumes. 

To define *limit superior*, we need calculus; you can think of this as just the *maximum*."=#

#= md"This conforms with our usual definiton of area in two dimensions.

But in $\mathbb{R}^2$ we can also calculate area more directly:" =#

# ╔═╡ bb9ef4d7-c589-4157-99ad-4d1bcb1e0e2b
md"For a 2-by-2 matrix this volume is the area of a parallelogram which we can compute by hand."

# ╔═╡ ac657e72-0e98-11eb-1a8c-b70f72979c6b
begin
	function enclosing_rectangle(A)
		u = A[1:2, 1]
		v = A[1:2, 2]
		w = u + v
	
		x1 = minimum([0 u[1] v[1] w[1]])
		x2 = maximum([0 u[1] v[1] w[1]])
	
		y1 = minimum([0 u[2] v[2] w[2]])
		y2 = maximum([0 u[2] v[2] w[2]])
	
		return x1, x2, y1, y2
	end

	x1, x2, y1, y2 = enclosing_rectangle(A)
	
	mplot = matrix_plot(A)
	plot!(Shape([x1, x2, x2, x1], [y1, y1, y2, y2]), color = :blue, opacity =.1)
end

# ╔═╡ 18b799c1-435e-427b-9be8-2e712d3c662c
# x-coordinates of points in picture
xcoords = 0, A[1,1], A[1,2], A[1,1] + A[1,2]

# ╔═╡ 9fe4f768-9d41-416c-9c1b-d8737c1a9a69
# y-coordinates of points in picture
ycoords = 0, A[2,1], A[2,2], A[2,1] + A[2,2]

# ╔═╡ 0139bd8b-57da-43df-9c51-9e81bbed4111
# outer rectangle area
outer_rectangle = (maximum(xcoords) - minimum(xcoords)) * (maximum(ycoords) - minimum(ycoords))

# ╔═╡ 744c9460-db89-4bef-9edc-9681c1d98d21
# outer triangle areas
outer_triangles = 4.22 * 2.28 + 0.98 * 4.72 + 2.28 * 0.98 * 2

# ╔═╡ b89360fe-614a-4e02-bbac-98f788b6b4d8
# remaining area
parallelogram_area = outer_rectangle - outer_triangles

# ╔═╡ 6413ac92-ab08-4003-b3ab-cbd34aad713b
md"This is exactly equal to the (absolute value) of the determinant of $A$:"

# ╔═╡ a3809b2e-f369-4ea7-8dec-4e9e8ce83a81
det(A)

# ╔═╡ e442a19c-2c34-4934-92dc-639a2e93716b
md"This turns out to be true of all square matrices: the volume of $A$ is $|\det(A)|$. 

Let's try to verify this property experimentally. 

There is a built-in function to compute $\det(A)$, but how can we estimate the volume of $A$ as defined above?"

# ╔═╡ e078d7c6-0ea3-11eb-307b-d1c904fb5539


# ╔═╡ e1faae26-0ea3-11eb-318d-77713a78456a
md"Here is a probabilistic definition of volume of an arbitrary region $S$ in $\mathbb{R}^n$ that we can use for this.

Choose $u \in \mathbb{R}^n$ and positive real numbers $L_1,L_2,\dots,L_n$ such that $S$ is contained in the $n$-dimensional box

$R = \{ u + v \in \mathbb{R}^n : 0\leq v_i \leq L_i\text{ for }i=1,2,\dots,n\}$

which has volume $L_1\cdot L_2\cdots L_n$.

Now sample many points uniformly at random from the region $R$. The *volume* of $S$ is then approximately the volume of $R$ times the fraction of these points that are contained in $S$. 

So if we generate $N$ points within $R$ and $m$ of them are inside $S$, then we expect 

$\mathrm{volume}(S) \approx \frac{m}{N} \cdot L_1\cdot L_2\cdots L_n.$

With enough random points $N$ this approximation  will converge to the true volume.

"

#=md"Here is an alternate, probabilistic definition of volume that is more amenable to estimation.

Consider a region $S$ in $\mathbb{R}^n$. Define the volume of 

$R = u + \mathrm{rect}(L_1,L_2,\dots,L_n) = \{ u + v \in \mathbb{R}^n : 0\leq v_i \leq L_i\text{ for }i=1,2,\dots,n\}$

to be $L_1\cdot L_2\cdots L_n$ as usual. Choose $u$ and $L_1,L_2,\dots,L_n$ such that

$S \subseteq R.$

Now sample many points uniformly at random from the region $R$. The *volume* of $S$ should then be (close to) the volume of $R$ times the fraction of these points that are contained in $S$. 

So if we generate $N$ points within $R$ and $m$ of them are inside $S$, then we expect 

$\mathrm{volume}(S) \approx \frac{m}{N} \cdot L_1\cdot L_2\cdots L_n.$

With enough random points $N$ this approximate volume will converge to the true volume.

"=#

# ╔═╡ fe034b28-0ea3-11eb-1b1a-1f474ab12faf


# ╔═╡ f921e1cc-0ea4-11eb-21cd-2305c9faf2d9
md"When our matrix $A$ is invertible and $S$ is the $n$-dimensional parallelogram associated to $A$, then there is an easy way for us to test whether a random point $w \in R$ belongs to $S$: 

We just check if $A^{-1} w$ has all coordinates between $0.0$ and $1.0$.

Let's write some code to implement this probabilistic algorithm to estimate the matrix volume, and then compare the result with the determinant."

# ╔═╡ 5e0bb100-0e49-11eb-272e-db4dce3e1696
function volume_estimates(A, lower_bound, upper_bound, trials)
	# we are trying to estimate the volume of the (n-dimension) parallelogram
	# whose edges include the vectors that are the columns of the matrix A
	#
	# we will sample points from a rectangular region R
	#
	# this outer region R is u + rect(L1, L2, ..., Ln) where
	#
	#   L1 = L2 = ... = Ln = upper_bound - lower_bound
	#   u = [lower_bound; lower_bound; ...; lower_bound]
	#
	
	(n, p) = size(A)
	@assert n == p
	
	estimates = []
	samples = []
	
	# let's only record the cumulative estimate every 1000 trials
	interval = maximum([Int(floor(trials/1000)) 1])
	
	# size of region R
	sample_region_size = (upper_bound - lower_bound)^n
	
	# rather than generate a single vector v many times,
	# we construct one random matrix with many columns
	v = rand(lower_bound:0.00001:upper_bound, n, trials)
	# now we can just do one matrix multiplication with all of our samples 
	w = A^-1 * v

	count = 0
	for j=1:trials
		# column j of v belongs to the matrix volume if and only if 
		# column j of w = A^-1 * v has all entries between 0.0 and 1.0
		if 0.0 <= minimum(w[:,j]) && maximum(w[:,j]) <= 1.0
			count += 1
		end
		
		if j % interval == 0
			fraction = count / j
			append!(estimates, fraction * sample_region_size)
			append!(samples, j)
		end
	end
	
	return estimates, samples
end

# ╔═╡ 9398ea44-0e4f-11eb-2412-23acd564dc41
function estimate_bounds(A, trials=1000)
	# we get more accurate volume estimates by using a smaller region R
	#
	# this method estimates optimal choices for lower_bound, upper_bound
	# so that R is as small as possible while containing the paralllelogram of A
	w = A * rand(0:1, size(A)[1], trials)
	return (minimum(w), maximum(w))
end

# ╔═╡ 7bdd6824-0e4a-11eb-063c-530768c6a7fe
function plot_estimates(matrix, lower, upper, trials)
	estimates, samples = volume_estimates(matrix, lower, upper, trials)
	plot(
		samples,
		estimates,
		xlabel="samples N",
		ylabel="volume estimate after N samples",
		#ylims = (0.0, 2 * estimates[end]),
		label="first estimate",
		title="Volume estimates of $(size(matrix)[1])-dimensional parallelogram",
		legend=:bottomright
	)
	
	estimates, samples = volume_estimates(matrix, lower, upper, trials)
	plot!(samples, estimates, label="second estimate")
	
	estimates, samples = volume_estimates(matrix, lower, upper, trials)
	plot!(samples, estimates, label="third estimate")
end

# ╔═╡ 7fb874de-0ea6-11eb-0449-39b8d954819a
trials = 20000

# ╔═╡ 843b9826-0ea6-11eb-3042-af3bcedae2ec
matrix = A

# ╔═╡ c5e50efa-0eb2-11eb-28dd-fdf089198c2d
lower, upper = estimate_bounds(matrix)

# ╔═╡ e21e9614-0ea6-11eb-2aba-97dd7bda343b
md"The following graph shows what happens when we sample a large number $N$ of points from a rectangular region $R$ containing the $n$-dimensional parallelogram $S$ of our matrix $A$, and then multiply the faction of the points that are in $S$ by the volume of $R$.

The resulting plots should converge to the volume of the parallelogram spanned by the columns $A$."

# ╔═╡ 1cc8a0d8-27fb-4be8-8c1e-f1257af6e217
plot_estimates(matrix, lower, upper, trials)

# ╔═╡ 7c4e9c3b-5776-4f2d-90ea-bd8b75fbb098
# here is what we computed for the exact area:
parallelogram_area

# ╔═╡ 18a01504-0ea6-11eb-1b64-c33c95d4bcae
# another way to compute this number (up to +/- sign):
det(A)

# ╔═╡ 0908992e-0ea6-11eb-211a-51a9655c77b8
md"Compare these values with our plots.

Moral: $\det(A)$ is as the *signed volume of the parallelogram spanned by the columns of $A$*.

We can observe this in any dimension, not just $n=2$.

However, in higher dimensions we need more samples to get a good estimate.
"

# ╔═╡ dd18e9c6-0eb3-11eb-31c4-296dbc11be15
M = rand(-5:0.001:5, 4, 4)

# ╔═╡ ece780c4-0eb3-11eb-1b24-0f804525f8ef
hlower, hupper = estimate_bounds(M)

# ╔═╡ c91999a2-0eb3-11eb-285a-85e605e2a1f6
htrials = 10000000

# ╔═╡ c4dabe7a-0eb3-11eb-306b-b3001b0c4f73
plot_estimates(M, hlower, hupper, htrials)

# ╔═╡ 00b1b865-4608-4589-8a3f-d83075c29b04
md"Our volume approximations shown in the graph above should converge to numbers that are close to the absolute value of the following determinant:"

# ╔═╡ 5645ac26-0eb4-11eb-3cfe-d13415915d87
# actual volume (up to +/- sign)
det(M)

# ╔═╡ ef65faf4-f273-4324-bedd-63a09adf8541


# ╔═╡ bbcc1392-0eb3-11eb-1868-b39c9b4130d4
md"""What about the sign of the determinant?

Some terminology: 
* An invertible matrix with positive determinant is called **orientation-preserving**.
* An invertible matrix with negative determinant is called **orientation-reversing**.

Whether an $n\times n$ matrix is orientation-preserving or orientation-reversing is something we can "just see," at least when $n=2$, simply by looking at how the matrix transforms a set of points. 
"""

# ╔═╡ c89d30ef-c2b0-4cde-ad08-4d020bf15990
md"A black and white image is encoded by a set of vectors $v = \left[\begin{array}{c} v_1 \\ v_2 \end{array}\right] \in \mathbb{R}^2$: one vector for each black pixel at horizontal position $v_1$ and vertical position $v_2$.

Suppose we have a $2\times 2$ matrix $A$.
Then we may use $A$ to transform our image to another image, in which each vector $v$ is replaced by $Av$. 

By comparing these images, we will be able to see whether $A$ is orientation-preserving or orientation-reversing."

# ╔═╡ 151ce91a-78bf-4430-8901-9bfa1ab889e8
function extract_image(filename)
	# this function converts a png file to a list of vectors as we just described
	img = download(filename) |> load
	gx, gy = [], []
	(m,n) = size(img)
	for x=1:m
		for y=1:n 
			if img[x,y] != img[1,1]
				append!(gx, x)
				append!(gy, y)
			end
		end
	end
	# now resize list of coordinates into 2-by-N matrix
	return vcat(transpose(reverse(gx)), transpose(gy))
	# note: we reverse order of x-coordinates because increasing the row
	# index goes down in a matrix, which is opposite of cartesian coordinates
end

# ╔═╡ bb5541e1-439e-4722-90c2-8b2a467a4fcd
g = extract_image("www.math.ust.hk/~emarberg/teaching/2020/Math2121/julia/base.png")

# ╔═╡ eda83f52-e556-4996-a409-f3825dbbd3de
md"Here is what this image looks like if we plot all of the points:"

# ╔═╡ 69487a04-b512-45cc-b9ce-42d10162484a
function make_displayable(g)
	# this function converts our image as list of vectors into something displayable
	padding = 3
	gx, gy = g[1,:], g[2,:]
	
	minx, miny = Int(floor(minimum(gx))) - 1, Int(floor(minimum(gy))) - 1
	maxx, maxy = Int(ceil(maximum(gx))), Int(ceil(maximum(gy)))
	
	ans = zeros(Int64, maxx - minx + padding, maxy - miny + padding)
	for i = 1:length(gx)
		x = Int(ceil(gx[i])) - minx
		y = Int(ceil(gy[i])) - miny
		for p=0:padding
			for q=0:padding
				ans[x + p, y + q] = 1
			end
		end
	end
	return ans
end

# ╔═╡ b7b92b30-6088-46a7-b769-bfc3d8540104
heatmap(
	make_displayable(g), 
	aspect_ratio=:equal, 
	legend=false, 
	color=:binary,
	title="Starting set in R^2"
)

# ╔═╡ ac418e32-600d-4756-b491-9d71fed37a0d
md"If $A$ is an $n\times n$ matrix with $\det(A) \neq 0$, then the matrix

$B = \frac{1}{|\det(A)|^{1/n}} A$

has $\det(B) = \pm 1$. 

If we transform an image using $B$ instead of $A$ (when $n=2$), then the resulting image will be the same, just rescaled to have the same size as the starting image. 

So to observe the qualitative difference between matrices with positive or negative determinant, we can just consider matrices with determinant $+1$ or $-1$."

# ╔═╡ 60db7888-dfe3-4f90-ad29-603332b7a0f0
md"Below are some functions to create random $n$-by-$n$ matrices with determinant $+1$ or $-1$.

We'll just need the $n=2$ case of these methods."

# ╔═╡ 4a862dc5-e329-47cd-928c-b9292a76c297
function randmat(n)
	# returns n-by-n matrices with uniformly random entries from interval [-1,1]
	return 2 * rand(Float64, n, n) .- 1.0
end

# ╔═╡ 1855a26e-daf7-481c-85a1-0ea14208dc1e
function positive_determinant(n)
	while true
		A = randmat(n)
		if det(A) < 0.1
			continue
		end
		return A / abs(det(A))^1/n
	end
end

# ╔═╡ c4e88642-d9a2-4757-a342-4c0ab1afb1e5
function negative_determinant(n)
	while true
		A = randmat(n)
		if det(A) > -0.1
			continue
		end
		return A / abs(det(A))^1/n
	end
end

# ╔═╡ 97e43a2b-b3f9-4ad0-ba83-0e826a18946e
function illustrate(g, A, B)
	# function to plot side-by-side our image g transformed by A and B
	plot(
		heatmap(
			make_displayable(A * g), 
			legend=false, 
			color=:binary
		),
		heatmap(
			make_displayable(B * g), 
			legend=false,
			color=:binary
		),
		layout = (1, 2),
	)
end		

# ╔═╡ 341dc2b4-195c-442c-b72c-eb99effe792a
md"""The plots below show our starting image transformed by random matrices $A$ with positive and negative determinant. 

If the matrix $A$ has positive determinant, then we say $A$ is **orientation-preserving** because the transformed image "looks the same (but rotated)." This is shown on the left.

If the matrix $A$ has negative determinant, then we say $A$ is **orientation-reversing** because the transformed image is "the mirror image (but rotated)" of what we started with. This is on the right."""

# ╔═╡ 1a85129b-9fdc-4d82-bb5b-8c0fd0c4b606
illustrate(g, positive_determinant(2), negative_determinant(2))

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version = "1.17.1+0"

[[deps.Xorg_libxkbfile_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Xorg_libX11_jll"]
git-tree-sha1 = "ed756a03e95fff88d8f738ebc2849431bdd4fd1a"
registries = "General"
uuid = "cc61e674-0454-545c-8b26-ed2c68acab7a"
version = "1.2.0+0"

[[deps.Xorg_xcb_util_image_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Xorg_xcb_util_jll"]
git-tree-sha1 = "f4fc02e384b74418679983a97385644b67e1263b"
registries = "General"
uuid = "12413925-8142-5f55-bb0e-6d7ca50bb09b"
version = "0.4.1+0"

[[deps.Xorg_xcb_util_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Xorg_libxcb_jll"]
git-tree-sha1 = "68da27247e7d8d8dafd1fcf0c3654ad6506f5f97"
registries = "General"
uuid = "2def613f-5ad1-5310-b15b-b15d46f528f5"
version = "0.4.1+0"

[[deps.Xorg_xcb_util_keysyms_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Xorg_xcb_util_jll"]
git-tree-sha1 = "44ec54b0e2acd408b0fb361e1e9244c60c9c3dd4"
registries = "General"
uuid = "975044d2-76e6-5fbe-bf08-97ce7c6574c7"
version = "0.4.1+0"

[[deps.Xorg_xcb_util_renderutil_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Xorg_xcb_util_jll"]
git-tree-sha1 = "5b0263b6d080716a02544c55fdff2c8d7f9a16a0"
registries = "General"
uuid = "0d47668e-0667-5a69-a72c-f761630bfb7e"
version = "0.3.10+0"

[[deps.Xorg_xcb_util_wm_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Xorg_xcb_util_jll"]
git-tree-sha1 = "f233c83cad1fa0e70b7771e0e21b061a116f2763"
registries = "General"
uuid = "c22f9ab0-d5fe-5066-847c-f4bb1cd4e361"
version = "0.4.2+0"

[[deps.Xorg_xkbcomp_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Xorg_libxkbfile_jll"]
git-tree-sha1 = "9a5e6e8f1a7c40941465c04840145e527f0d6b4b"
registries = "General"
uuid = "35661453-b289-5fab-8a00-3d9160c6a3a4"
version = "1.5.0+0"

[[deps.Xorg_xkeyboard_config_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Xorg_xkbcomp_jll"]
git-tree-sha1 = "df77b22eb77051b3d0ef30aab979684f7482c66f"
registries = "General"
uuid = "33bec58e-1273-512f-9401-5d533626f822"
version = "2.48.0+0"

[[deps.Xorg_xtrans_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl"]
git-tree-sha1 = "a63799ff68005991f9d9491b6e95bd3478d783cb"
registries = "General"
uuid = "c5fb5394-a638-5e4d-96e5-b29de1b5cf10"
version = "1.6.0+0"

[[deps.Zlib_jll]]
deps = ["Libdl"]
uuid = "83775a58-1f1d-513f-b197-d71354ab007a"
version = "1.3.1+2"

[[deps.Zstd_jll]]
deps = ["CompilerSupportLibraries_jll", "Libdl"]
uuid = "3161d3a3-bdf6-5164-811a-617609db77b4"
version = "1.5.7+1"

[[deps.fzf_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl"]
git-tree-sha1 = "b6a34e0e0960190ac2a4363a1bd003504772d631"
registries = "General"
uuid = "214eeab7-80f7-51ab-84ad-2988db7cef09"
version = "0.61.1+0"

[[deps.libaom_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl"]
git-tree-sha1 = "1210ba774d3427387d307bf1f416d699b7c39417"
registries = "General"
uuid = "a4ae2306-e953-59d6-aa16-d00cac43593b"
version = "3.15.1+0"

[[deps.libass_jll]]
deps = ["Artifacts", "Bzip2_jll", "FreeType2_jll", "FriBidi_jll", "HarfBuzz_jll", "JLLWrappers", "Libdl", "Zlib_jll"]
git-tree-sha1 = "cb007192783c56d8249db4cf0e3495001edfe414"
registries = "General"
uuid = "0ac62f75-1d6f-5e53-bd7c-93b484bb37c0"
version = "0.17.5+0"

[[deps.libblastrampoline_jll]]
deps = ["Artifacts", "Libdl"]
uuid = "8e850b90-86db-534c-a0d3-1478176c7d93"
version = "5.15.0+0"

[[deps.libdecor_jll]]
deps = ["Artifacts", "Dbus_jll", "JLLWrappers", "Libdl", "Libglvnd_jll", "Pango_jll", "Wayland_jll", "xkbcommon_jll"]
git-tree-sha1 = "9bf7903af251d2050b467f76bdbe57ce541f7f4f"
registries = "General"
uuid = "1183f4f0-6f2a-5f1a-908b-139f9cdfea6f"
version = "0.2.2+0"

[[deps.libdrm_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Xorg_libpciaccess_jll"]
git-tree-sha1 = "28e57478e8a160d346a19c28b3fffb9273bcc9c2"
registries = "General"
uuid = "8e53e030-5e6c-5a89-a30b-be5b7263a166"
version = "2.4.134+0"

[[deps.libfdk_aac_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl"]
git-tree-sha1 = "646634dd19587a56ee2f1199563ec056c5f228df"
registries = "General"
uuid = "f638f0a6-7fb0-5443-88ba-1cc74229b280"
version = "2.0.4+0"

[[deps.libpng_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Zlib_jll"]
git-tree-sha1 = "1fcd6cf05d23adab0b5ffc496d020537215751dc"
registries = "General"
uuid = "b53b4c65-9356-5827-b1ea-8c7a1a84506f"
version = "1.6.59+1"

[[deps.libsixel_jll]]
deps = ["Artifacts", "JLLWrappers", "JpegTurbo_jll", "Libdl", "libpng_jll"]
git-tree-sha1 = "e067c8bae65bb40866552296a2d5b4dc65b8928f"
registries = "General"
uuid = "075b6546-f08a-558a-be8f-8157d0f608a5"
version = "1.80.702+0"

[[deps.libva_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Xorg_libX11_jll", "Xorg_libXext_jll", "Xorg_libXfixes_jll", "libdrm_jll"]
git-tree-sha1 = "7dbf96baae3310fe2fa0df0ccbb3c6288d5816c9"
registries = "General"
uuid = "9a156e7d-b971-5f62-b2c9-67348b8fb97c"
version = "2.23.0+0"

[[deps.libvorbis_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Ogg_jll"]
git-tree-sha1 = "11e1772e7f3cc987e9d3de991dd4f6b2602663a5"
registries = "General"
uuid = "f27f6e37-5d2b-51aa-960f-b287f2bc3b7a"
version = "1.3.8+0"

[[deps.libwebp_jll]]
deps = ["Artifacts", "Giflib_jll", "JLLWrappers", "JpegTurbo_jll", "Libdl", "Libglvnd_jll", "Libtiff_jll", "libpng_jll"]
git-tree-sha1 = "ccbb625a89ec6195856a50aa2b668a5c08712c94"
registries = "General"
uuid = "c5f90fcd-3b7e-5836-afba-fc50a0988cb2"
version = "1.4.0+0"

[[deps.nghttp2_jll]]
deps = ["Artifacts", "CompilerSupportLibraries_jll", "Libdl"]
uuid = "8e850ede-7688-5339-a07c-302acd2aaf8d"
version = "1.67.1+0"

[[deps.oneTBB_jll]]
deps = ["Artifacts", "JLLWrappers", "LazyArtifacts", "Libdl"]
git-tree-sha1 = "ed2d8d4ad96c03d3abe3382cdbbdea9489b63636"
registries = "General"
uuid = "1317d2d5-d96f-522e-a858-c73665f53c3e"
version = "2023.1.0+0"

[[deps.p7zip_jll]]
deps = ["Artifacts", "CompilerSupportLibraries_jll", "Libdl"]
uuid = "3f19e933-33d8-53b3-aaab-bd5110c3b7a0"
version = "17.8.2+0"

[[deps.x264_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl"]
git-tree-sha1 = "14cc7083fc6dff3cc44f2bc435ee96d06ed79aa7"
registries = "General"
uuid = "1270edf5-f2f9-52d2-97e9-ab00b5d0237a"
version = "10164.0.1+0"

[[deps.x265_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl"]
git-tree-sha1 = "e7b67590c14d487e734dcb925924c5dc43ec85f3"
registries = "General"
uuid = "dfaa095f-4041-5dcd-9319-2fabd8486b76"
version = "4.1.0+0"

[[deps.xkbcommon_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Xorg_libxcb_jll", "Xorg_xkeyboard_config_jll"]
git-tree-sha1 = "a1fc6507a40bf504527d0d4067d718f8e179b2b8"
registries = "General"
uuid = "d8fb68d0-12a3-5cfd-a85a-d49703b185fd"
version = "1.13.0+0"

[registries.General]
url = "https://github.com/JuliaRegistries/General.git"
uuid = "23338594-aafe-5451-b93e-139f81909106"
"""

# ╔═╡ Cell order:
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# ╠═4a862dc5-e329-47cd-928c-b9292a76c297
# ╠═1855a26e-daf7-481c-85a1-0ea14208dc1e
# ╠═c4e88642-d9a2-4757-a342-4c0ab1afb1e5
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# ╟─341dc2b4-195c-442c-b72c-eb99effe792a
# ╠═1a85129b-9fdc-4d82-bb5b-8c0fd0c4b606
# ╟─00000000-0000-0000-0000-000000000001
# ╟─00000000-0000-0000-0000-000000000002