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Your currently chosen function is:

$$f_1(x) = (x^3 - 1) / x$$

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2D Functions

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RGB to HSV

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🌞 For this reason, if we want to ensure that our colors are uniformely distributed in the base domain, we need to rescale the magnitudes logarithmically. One such way to do this is using our $g_1$ function.

👇 But you can also set your own function! Try it out, change this to a different expression and see how both the base and color domains change:

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Try!

Type anything, try it out!

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Your chosen point:

x = -0.04 + 0.72im

Your function value:

f(x) = -0.44 + 1.327im

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That's pretty boring and unintuitive to understand right? Isn't there a better way?

Well there is: let's use colors! 🎨

Now each point in our initial space has a unique position and a unique color. So we follow these steps:

👉 We choose our point (or color) $x$ 🔴

👉 Calculate our function value $f(x)$

👉 Find the corresponding color for that value in the initial space 🔵

✨ Try it out!

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Tranformation functions

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Your chosen point: x = -0.04 + 0.72im

Your function value: f(x) = -0.44 + 1.327im

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Math Art 📏🎨

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👇And now let's choose again a function

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Number Picker from PlutoImageCoordinatePicker

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You (hopefully) know from school, that we can define a function for a number, such as $f(x) = 2x$ which just basically means for any number $x$ ❗ we want our function $f$ to double ‼️ that value. So functions are nothing more than just a transformation of the numbers! There are countless functions in math, in fact they kind of define the very basics of mathematics 😉

👇 Start by chosing a function $f$ below:

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🎨 Domain Coloring

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Now if you're like me (or like most people), you probably didn't think of the word 'Art' first, because often we don't associate art🎨 with mathematics📏. And yet today we're going to see how we can create some beautiful (slightly trippy🌀) art by using only math.

Are you ready? 💃

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Let's start by choosing our initial drawing, or what we call domain. We're not limited to any specific respresentation. We can give each number any color we want as long as each one gets its unique color.

👇 Try it out! Choose a different domain color

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Great! Now let's choose our point $x$. To make thing more fun, we'll use complex numbers. If you don't know what that is, that's okay, for now you can think of it as a 2D point

👇 Click anywhere to chose a point!

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Coloring and Formating

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Hey there! Let's play a round of Word Link, I will say a word and you have to write down the first word that comes to mind in response:

What's the first you can think of when I say 'Math'?

👇 Type your answer below:

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$$f_5(x) = x^9 - 1/ x^3 + 1$$

text/htmlf2text/plain٠

$$f_2(x) = \frac{x - \frac{1}{2} (1 + i)}{x^2}$$

text/htmlf4text/plainW

$$f_4(x) = sin(x)$$

text/htmlf1text/plain^

$$f_1(x) = (x^3 - 1) / x$$

text/htmlf3text/plainL

$$f_3(x) = 1/x$$

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Advanced: Functions and Code

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Your currently chosen function is:

$$f_1(x) = (x^3 - 1) / x$$

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Your turn 🫵

Now it's your turn: Let your creativity run free!! Here you can choose your own domain coloring 🎨 and own transformation 📏 and create all the awesome art you want! 🤪

Have fun 🥳

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Much better, right?! Now, we're ready for some art! 🎉

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Functions for initial domain

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Pro Tip: The base function

⚠️ Notice that if we were to really plot the identify function, we would get a big black square because the density of the points in polar coordinates is not uniform.

👇 Try it out: Replace custom_g with the identity function

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And tadaaa 🎉

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To finally create our art pieces, all we have to do is repeat the same thing we did so far for all the points in our 2D space. So instead of checking the values for each point one by one, we will do the following:

👉 Go over each point in the initial drawing one by one 🔵

👉 Apply the transformation from above 📏

👉 Put in the transformed color 🔴 in the place of the original one in the new drawing

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nbpkginstall_time_nsQyinstantiatedòinstalled_versionsImageIO0.6.9PlutoUI0.7.81!__internal_julia_manifest_version1.12.4__internal_julia_version1.12.4PlutoImageCoordinatePicker1.4.2Colors0.13.1ImageShow0.3.8PlutoTeachingTools0.4.7terminal_outputsnbpkg_sync& Waiting for other notebooks to finish Pkg operations... === Resolving... ===  Project No packages added to or removed from `~/.julia/scratchspaces/c3e4b0f8-55cb-11ea-2926-15256bba5781/pkg_envs/env_oyuesckvoh/Project.toml`  Manifest No packages added to or removed from `~/.julia/scratchspaces/c3e4b0f8-55cb-11ea-2926-15256bba5781/pkg_envs/env_oyuesckvoh/Manifest.toml` Instantiating... === Precompiling... ===ImageIO& Waiting for other notebooks to finish Pkg operations... === Resolving... ===  Project No packages added to or removed from `~/.julia/scratchspaces/c3e4b0f8-55cb-11ea-2926-15256bba5781/pkg_envs/env_oyuesckvoh/Project.toml`  Manifest No packages added to or removed from `~/.julia/scratchspaces/c3e4b0f8-55cb-11ea-2926-15256bba5781/pkg_envs/env_oyuesckvoh/Manifest.toml` Instantiating... === Precompiling... ===PlutoUI& Waiting for other notebooks to finish Pkg operations... === Resolving... ===  Project No packages added to or removed from `~/.julia/scratchspaces/c3e4b0f8-55cb-11ea-2926-15256bba5781/pkg_envs/env_oyuesckvoh/Project.toml`  Manifest No packages added to or removed from `~/.julia/scratchspaces/c3e4b0f8-55cb-11ea-2926-15256bba5781/pkg_envs/env_oyuesckvoh/Manifest.toml` Instantiating... === Precompiling... ===PlutoImageCoordinatePicker& Waiting for other notebooks to finish Pkg operations... === Resolving... ===  Project No packages added to or removed from `~/.julia/scratchspaces/c3e4b0f8-55cb-11ea-2926-15256bba5781/pkg_envs/env_oyuesckvoh/Project.toml`  Manifest No packages added to or removed from `~/.julia/scratchspaces/c3e4b0f8-55cb-11ea-2926-15256bba5781/pkg_envs/env_oyuesckvoh/Manifest.toml` Instantiating... === Precompiling... ===Colors& Waiting for other notebooks to finish Pkg operations... === Resolving... ===  Project No packages added to or removed from `~/.julia/scratchspaces/c3e4b0f8-55cb-11ea-2926-15256bba5781/pkg_envs/env_oyuesckvoh/Project.toml`  Manifest No packages added to or removed from `~/.julia/scratchspaces/c3e4b0f8-55cb-11ea-2926-15256bba5781/pkg_envs/env_oyuesckvoh/Manifest.toml` Instantiating... === Precompiling... ===PlutoTeachingTools& Waiting for other notebooks to finish Pkg operations... === Resolving... ===  Project No packages added to or removed from `~/.julia/scratchspaces/c3e4b0f8-55cb-11ea-2926-15256bba5781/pkg_envs/env_oyuesckvoh/Project.toml`  Manifest No packages added to or removed from `~/.julia/scratchspaces/c3e4b0f8-55cb-11ea-2926-15256bba5781/pkg_envs/env_oyuesckvoh/Manifest.toml` Instantiating... === Precompiling... ===ImageShow& Waiting for other notebooks to finish Pkg operations... === Resolving... ===  Project No packages added to or removed from `~/.julia/scratchspaces/c3e4b0f8-55cb-11ea-2926-15256bba5781/pkg_envs/env_oyuesckvoh/Project.toml`  Manifest No packages added to or removed from `~/.julia/scratchspaces/c3e4b0f8-55cb-11ea-2926-15256bba5781/pkg_envs/env_oyuesckvoh/Manifest.toml` Instantiating... === Precompiling... ===enabled÷restart_recommended_msgrestart_required_msgbusy_packageswaiting_for_permission,waiting_for_permission_but_probably_disabled«cell_inputsB$b9616798-709e-417a-b850-a1d0fc4cf61acell_id$b9616798-709e-417a-b850-a1d0fc4cf61acode;md"Your currently chosen function is: $(function_names[f])"metadatashow_logsèdisabled®skip_as_script«code_folded$b07e51fd-b4e1-4baf-bf08-5e475f7ef744cell_id$b07e51fd-b4e1-4baf-bf08-5e475f7ef744codemd"## 2D Functions"metadatashow_logsèdisabled®skip_as_script«code_folded$64a04229-067d-448d-866e-15fae2938420cell_id$64a04229-067d-448d-866e-15fae2938420codemd"### RGB to HSV"metadatashow_logsèdisabled®skip_as_script«code_folded$32701e29-5a09-4ef5-b2f3-9be3e16fe411cell_id$32701e29-5a09-4ef5-b2f3-9be3e16fe411codepmd""" 🌞 For this reason, if we want to ensure that our colors are uniformely distributed in the base domain, we need to rescale the magnitudes logarithmically. One such way to do this is using our $g_1$ function. 👇 But you can also set your own function! Try it out, change this to a different expression and see how both the base and color domains change: """metadatashow_logsèdisabled®skip_as_script«code_folded$ca565680-2c1b-4a68-8877-0df94c7fe0a6cell_id$ca565680-2c1b-4a68-8877-0df94c7fe0a6codefunction ComplexNumberPicker(axes; default::Union{Real,Complex,Nothing}=nothing) t(x) = (x - 150.0) / 120.0 tinv(x) = x * 120.0 + 150.0 default_coord = default === nothing ? nothing : PlutoImageCoordinatePicker.ClickCoordinate(300, 300, tinv(real(default)), tinv(-imag(default)) ) PlutoUI.Experimental.transformed_value(ImageCoordinatePicker(axes; default=default_coord)) do point if point === nothing nothing else t(point.x) - im * t(point.y) end end endmetadatashow_logsèdisabled®skip_as_script«code_folded$9fb31e84-d4e1-4b37-ab4a-17f2dab7a5fecell_id$9fb31e84-d4e1-4b37-ab4a-17f2dab7a5fecodefunction_choosermetadatashow_logsèdisabled®skip_as_script«code_folded$198332aa-1070-4cbc-88a3-af0d26ff7f2ecell_id$198332aa-1070-4cbc-88a3-af0d26ff7f2ecodebegin if answer == "" md""" !!! correct "Try!" Type anything, try it out!""" elseif answer == "Art" || answer == "art" md""" !!! correct "Impressive!" Now that's unexpected, great job! 🎉 I really want to know what kind of math you were taught at school, sounds more fun than the one I had!""" else md""" !!! correct "Nice!" There's really no correct answer here, but it's nice to see how your brain works ☺️ Fun fact: the most common words are: *Numbers*, *Equations*, *Algebra* """ end endmetadatashow_logsèdisabled®skip_as_script«code_folded$9ae3fe47-4c53-48c1-a126-bad1d1615f01cell_id$9ae3fe47-4c53-48c1-a126-bad1d1615f01code9function g1(r) return (1.0 - 1.0 / (1 + r^2))^(0.2) end metadatashow_logsèdisabled®skip_as_script«code_folded$e59508e9-9254-48eb-8c5f-5de361c903dfcell_id$e59508e9-9254-48eb-8c5f-5de361c903dfcodePlutoTeachingTools.Columns( # Left c_picker_colored, # Right let md""" **Your chosen point:** $x_color x = $x_col_trimed **Your function value:** $fx_color f(x) = $fx_colored""" end )metadatashow_logsèdisabled®skip_as_script«code_folded$caee3573-b2cf-4ff4-b8f5-a1ddff1c4718cell_id$caee3573-b2cf-4ff4-b8f5-a1ddff1c4718codemd"That's pretty boring and unintuitive to understand right? Isn't there a better way? Well there is: let's use colors! 🎨 Now each point in our initial space has a *unique* position **and** a *unique* color. So we follow these steps: 👉 We choose our point (or color) $x$ 🔴 👉 Calculate our function value $f(x)$ 👉 Find the corresponding **color** for that value in the initial space 🔵 ✨ Try it out!"metadatashow_logsèdisabled®skip_as_script«code_folded$4a583be7-7ad9-4bc9-9788-ff5573eaac69cell_id$4a583be7-7ad9-4bc9-9788-ff5573eaac69codemd"### Tranformation functions"metadatashow_logsèdisabled®skip_as_script«code_folded$5f7297f5-2412-4576-ae35-4bda3004d1d9cell_id$5f7297f5-2412-4576-ae35-4bda3004d1d9codeLc_fractal_picker = @bind x ComplexNumberPicker(im_axes, default=-.04+.72im);metadatashow_logsèdisabled®skip_as_script«code_folded$1948ca3e-619b-4a4f-938d-97d3d9b5de65cell_id$1948ca3e-619b-4a4f-938d-97d3d9b5de65code# Returns the color corresponding to one point function color_value(x, g; v::Real=1.0) Z = x # Compute H (hue), S (saturation), V (value) H = rad2deg.(angle.(Z)) # Convert to degrees angle S = v # Constant value V = g.(abs.(Z)) # Combine H, S, V into an HSV image hsv_array = reshape(hcat(H, S, V), 1, 1, 3) # Convert HSV image to RGB image height, width = size(hsv_array) rgb_image = Array{RGB, 2}(undef, height, width) color = HSV(H, S, V) rgb = RGB(color) return rgb endmetadatashow_logsèdisabled®skip_as_script«code_folded$d07c146a-e106-42fb-9647-6e4132e51740cell_id$d07c146a-e106-42fb-9647-6e4132e51740code\c_picker_colored = @bind x_colored ComplexNumberPicker(standard_domain, default=-.04+.72im);metadatashow_logsèdisabled®skip_as_script«code_folded$44473d2a-a38b-449c-a739-6ac7025c6c27cell_id$44473d2a-a38b-449c-a739-6ac7025c6c27code# Trims the c value in the picker to 3 digits after the , function trim_c(c) # Extract real and imaginary parts real_part = real(c) imag_part = imag(c) # Trim the real and imaginary parts real_part_trimmed = round(real_part, digits=3) # Round to 3 decimal places imag_part_trimmed = round(imag_part, digits=3) # Combine trimmed parts back into a complex number trimmed_num = real_part_trimmed + imag_part_trimmed * im return trimmed_num endmetadatashow_logsèdisabled®skip_as_script«code_folded$cfa96c23-a980-416c-a11d-d03b0d476984cell_id$cfa96c23-a980-416c-a11d-d03b0d476984codeYusing PlutoUI, PlutoTeachingTools, Colors, ImageShow, ImageIO, PlutoImageCoordinatePickermetadatashow_logsèdisabled®skip_as_script«code_folded$79d23ca2-0034-4b04-997a-14a29e4c2df6cell_id$79d23ca2-0034-4b04-997a-14a29e4c2df6code1@bind g Select([g1 => "Standard", g2 => "Fancy"])metadatashow_logsèdisabled®skip_as_script«code_folded$5abc3c95-4cad-4671-8b29-1faa8bc81919cell_id$5abc3c95-4cad-4671-8b29-1faa8bc81919codeDaside(md"👈 Set your transformation function here", v_offset=-455)metadatashow_logsèdisabled®skip_as_script«code_folded$612e7826-aae1-41eb-a507-4d7cd2c9f586cell_id$612e7826-aae1-41eb-a507-4d7cd2c9f586code٣PlutoTeachingTools.Columns( # Left c_fractal_picker, # Right let md""" **Your chosen point:** x = $x_trim **Your function value:** f(x) = $fx""" end )metadatashow_logsèdisabled®skip_as_script«code_folded$e5c408b1-8285-4898-988c-731ad1a59ecccell_id$e5c408b1-8285-4898-988c-731ad1a59ecccodeٔfunction_chooser = @bind f Select([f1 => md"Cool f1", f2 => "Super cool f2", f3 => "Boring Inverse f3", f4 => "Fun sin f4", f5 => "Wild function"]);metadatashow_logsèdisabled®skip_as_script«code_folded$261f1a0d-a261-42fb-9c57-ea0ba1846282cell_id$261f1a0d-a261-42fb-9c57-ea0ba1846282codemd"## Math Art 📏🎨"metadatashow_logsèdisabled®skip_as_script«code_folded$28c92348-5d69-47cb-ace3-f5ca04447237cell_id$28c92348-5d69-47cb-ace3-f5ca04447237code/md" 👇And now let's choose again a function "metadatashow_logsèdisabled®skip_as_script«code_folded$b0fa6891-4626-4269-81c6-b118c8d41c48cell_id$b0fa6891-4626-4269-81c6-b118c8d41c48code١aside(md""" !!! tip "💡 Tip" You can mix around and combine different types of functions: add, substract, multiply them together. Go wild! """, v_offset=-400)metadatashow_logsèdisabled®skip_as_script«code_folded$9b72deef-6212-4b67-a314-8be3a983a73acell_id$9b72deef-6212-4b67-a314-8be3a983a73acodeكbegin struct SVG content end function Base.show(io::IO, m::MIME"image/svg+xml", s::SVG) write(io, s.content) end SVG endmetadatashow_logsèdisabled®skip_as_script«code_folded$de52fc27-a144-4ad2-8c47-4444b5abbe0acell_id$de52fc27-a144-4ad2-8c47-4444b5abbe0acode&custom_g(x) = x # g1(x) # Replace thismetadatashow_logsèdisabled®skip_as_script«code_folded$7e231820-c804-4581-8dcf-b5f9824931b3cell_id$7e231820-c804-4581-8dcf-b5f9824931b3code7md"### Number Picker from `PlutoImageCoordinatePicker`"metadatashow_logsèdisabled®skip_as_script«code_folded$f9efc61e-3fc9-48a0-85c8-58f24550e308cell_id$f9efc61e-3fc9-48a0-85c8-58f24550e308codef4(z) = sin(z)metadatashow_logsèdisabled®skip_as_script«code_folded$ba4f4620-f98f-41e3-bfb5-9e704a93d98ecell_id$ba4f4620-f98f-41e3-bfb5-9e704a93d98ecode8const im_axes = """ Real Imaginary 0 1 -1 i -i """ |> SVGmetadatashow_logsèdisabled®skip_as_script«code_folded$b27a9694-bf59-4f61-b527-2d54e0a28129cell_id$b27a9694-bf59-4f61-b527-2d54e0a28129codemd"You (hopefully) know from school, that we can define a function for a number, such as $f(x) = 2x$ which just basically means for any number $x$ ❗ we want our function $f$ to double ‼️ that value. So functions are nothing more than just a **transformation** of the numbers! There are countless functions in math, in fact they kind of define the very basics of mathematics 😉 👇 Start by chosing a function $f$ below:"metadatashow_logsèdisabled®skip_as_script«code_folded$8803ea22-6cdc-4e5b-85b3-f89a70f2e627cell_id$8803ea22-6cdc-4e5b-85b3-f89a70f2e627code# Calculates the colors for the whole domain function color_domain(f::Function, g::Function; v::Real=1.0) # Create complex grid z = x + 1im*y (NICE JULIA WAY TO DO THIS) depth = 300 x = range(-1.25,1.25,length=depth) y = range(-1.25,1.25,length=depth) grid = x' .- 1im .* y Z = f.(grid) # Compute H (hue), S (saturation), V (value) H = rad2deg.(angle.(Z)) # Convert to degrees angle S = v * ones(size(H)) # Constant value V = g.(abs.(Z)) # Combine H, S, V into an HSV image hsv_array = reshape(hcat(H, S, V), depth, depth, 3) # Convert HSV image to RGB image height, width = size(hsv_array) rgb_image = Array{RGB, 2}(undef, height, width) for i in 1:height for j in 1:width hue = hsv_array[i, j, 1] saturation = hsv_array[i, j, 2] value = hsv_array[i, j, 3] color = HSV(hue, saturation, value) rgb_image[i, j] = RGB(color) end end return rgb_image endmetadatashow_logsèdisabled®skip_as_script«code_folded$48380e35-1985-47a1-aa4d-1da27837839ccell_id$48380e35-1985-47a1-aa4d-1da27837839ccodemd"# 🎨 Domain Coloring"metadatashow_logsèdisabled®skip_as_script«code_folded$997932b8-2a6e-4ef7-8c91-2b1fc2d92289cell_id$997932b8-2a6e-4ef7-8c91-2b1fc2d92289code,md"Now if you're like me (or like most people), you probably didn't think of the word 'Art' first, because often we don't associate art🎨 with mathematics📏. And yet today we're going to see how we can create some beautiful (slightly trippy🌀) art by using **only** math. Are you ready? 💃"metadatashow_logsèdisabled®skip_as_script«code_folded$91758932-ef9e-4265-b32b-1edb4c0a61d6cell_id$91758932-ef9e-4265-b32b-1edb4c0a61d6code md"Let's start by choosing our initial drawing, or what we call **domain**. We're not limited to any specific respresentation. We can give each number any color we want as long as each one gets its **unique** color. 👇 Try it out! Choose a different domain color"metadatashow_logsèdisabled®skip_as_script«code_folded$7c477f2a-691b-431d-9930-f5eb925ad356cell_id$7c477f2a-691b-431d-9930-f5eb925ad356codemd"Great! Now let's choose our point $x$. To make thing more fun, we'll use complex numbers. If you don't know what that is, that's okay, for now you can think of it as a **2D point** 👇 Click anywhere to chose a point!"metadatashow_logsèdisabled®skip_as_script«code_folded$98e9021c-fdf0-48d0-a44a-583f57256382cell_id$98e9021c-fdf0-48d0-a44a-583f57256382code/begin x_trim = trim_c(x) fx = trim_c(f(x)) end;metadatashow_logsèdisabled®skip_as_script«code_folded$7780f241-b477-45ab-8287-1cd6b5409e13cell_id$7780f241-b477-45ab-8287-1cd6b5409e13codef5(z) = z^9 - 1/ z^3 + 1metadatashow_logsèdisabled®skip_as_script«code_folded$a278c9af-3409-4bd2-b977-d21ef52aacc5cell_id$a278c9af-3409-4bd2-b977-d21ef52aacc5codemd"### Coloring and Formating"metadatashow_logsèdisabled®skip_as_script«code_folded$d8177c39-6490-441a-aa3d-f9bb1cee5ee0cell_id$d8177c39-6490-441a-aa3d-f9bb1cee5ee0codeaside(md""" !!! info "Functions 🤓" Example functions to use: 📈 Polynomials and fractions: $x^2$ or $\frac{1}{x}$ 📈 Trigonometric functions: $\sin$, $\cos$, $\tan$ 📈 Exponentials and logarithms: $2^x$, $e^2$, $\log(x)$""", v_offset=-160)metadatashow_logsèdisabled®skip_as_script«code_folded$cc3d1dfe-e167-4c8c-a276-9ccc079cae45cell_id$cc3d1dfe-e167-4c8c-a276-9ccc079cae45codemd"Hey there! Let's play a round of **Word Link**, I will say a word and you have to write down the first word that comes to mind in response: What's the first you can think of when I say **'Math'**? 👇 Type your answer below:"metadatashow_logsèdisabled®skip_as_script«code_folded$3efa98cd-1b19-464e-8923-191e01b8d659cell_id$3efa98cd-1b19-464e-8923-191e01b8d659codeٛbegin x_col_trimed = trim_c(x_colored); x_color = color_value(x_colored, g) fx_colored = trim_c(f(x_colored)); fx_color = color_value(fx_colored, g) end;metadatashow_logsèdisabled®skip_as_script«code_folded$ad915710-ef64-4aed-8514-f3214491b9abcell_id$ad915710-ef64-4aed-8514-f3214491b9abcode# TODO: Probably a better more automated way to do this function_names = Dict( f1 => md"$f_1(x) = (x^3 - 1) / x$", f2 => md"$f_2(x) = \frac{x - \frac{1}{2} (1 + i)}{x^2}$", f3 => md"$f_3(x) = 1/x$", f4 => md"$f_4(x) = sin(x)$", f5 => md"$f_5(x) = x^9 - 1/ x^3 + 1$", )metadatashow_logsèdisabled®skip_as_script«code_folded$525a91e4-9830-47ca-86dc-4bf0786b5df1cell_id$525a91e4-9830-47ca-86dc-4bf0786b5df1code_intensity_picker_circle = @bind v_circle PlutoUI.Slider(0:0.05:1, default=0.9, show_value=true)metadatashow_logsèdisabled®skip_as_script«code_folded$a254f5c8-110a-4d11-9737-9e350a5e6a81cell_id$a254f5c8-110a-4d11-9737-9e350a5e6a81codeٛlet domain = heatmap(color_domain(f0, custom_g)) rgb_image = heatmap(color_domain(custom_f, custom_g)) PlutoTeachingTools.Columns(domain, rgb_image) endmetadatashow_logsèdisabled®skip_as_script«code_folded$0fe41744-2ea3-4d95-a094-adffb6a9db17cell_id$0fe41744-2ea3-4d95-a094-adffb6a9db17code@bind answer TextField()metadatashow_logsèdisabled®skip_as_script«code_folded$90cd3a2f-1b66-410d-a4ef-4f0b3fac5047cell_id$90cd3a2f-1b66-410d-a4ef-4f0b3fac5047code#md"## Advanced: Functions and Code"metadatashow_logsèdisabled®skip_as_script«code_folded$008fd9ac-c5b5-4f0c-bf44-3c0368e0d21dcell_id$008fd9ac-c5b5-4f0c-bf44-3c0368e0d21dcode٭begin domain = heatmap(color_domain(f0, g; v)) # f0 is the identity functionn rgb_image = heatmap(color_domain(f, g; v)) PlutoTeachingTools.Columns(domain, rgb_image) endmetadatashow_logsèdisabled®skip_as_script«code_folded$a8f1c7a9-3a9a-4ff5-9203-b2b7bac74dd2cell_id$a8f1c7a9-3a9a-4ff5-9203-b2b7bac74dd2code;md"Your currently chosen function is: $(function_names[f])"metadatashow_logsèdisabled®skip_as_script«code_folded$a4f5dd94-acd3-4099-ab05-2e7f1e435ba3cell_id$a4f5dd94-acd3-4099-ab05-2e7f1e435ba3codef0(z) = zmetadatashow_logsèdisabled®skip_as_script«code_folded$02a36bc7-a969-4d18-b742-2daa8076fff4cell_id$02a36bc7-a969-4d18-b742-2daa8076fff4codemd"## Your turn 🫵 Now it's your turn: Let your creativity run free!! Here you can choose your own domain coloring 🎨 and own transformation 📏 and create all the awesome art you want! 🤪 Have fun 🥳"metadatashow_logsèdisabled®skip_as_script«code_folded$ba68a796-f347-4c80-a9f7-5a5e6d2ddc49cell_id$ba68a796-f347-4c80-a9f7-5a5e6d2ddc49code (r -> isnan(r) ? 0.0 : r) return (logm - floor(logm))^0.3 endmetadatashow_logsèdisabled®skip_as_script«code_folded$1ce4351a-e9fa-4e49-924b-b9d5146c20afcell_id$1ce4351a-e9fa-4e49-924b-b9d5146c20afcode%md"### Functions for initial domain "metadatashow_logsèdisabled®skip_as_script«code_folded$80ade766-5d49-11ef-200e-c99f313ca7bccell_id$80ade766-5d49-11ef-200e-c99f313ca7bccodeClet n = 500 # Size of image img = Array{RGB{Float64}, 2}(undef, n, n) # Initialize array # Define the center of the circle center = (n / 2, n / 2) radius = n / 2 # Fill the image for x in 1:n for y in 1:n # Coordinates tcx = (x - center[1]) / radius tcy = (y - center[2]) / radius distance = sqrt(tcx^2 + tcy^2) # Check if the point is within the circle if distance <= 1.0 # Get the hue angle (in degrees) hue = atan(tcy, tcx) * 180 / π hue = hue < 0 ? hue + 360 : hue # Ensure hue is in [0, 360] # Calculate saturation based on the distance from the center saturation = distance # Get value using the slider value = v_circle # Convert HSV to RGB color = HSV(hue, saturation, value) # Set the pixel color img[x, y] = RGB(color) else #Outside the circle, set the pixel to white img[x, y] = RGB(1, 1, 1) end end end # Show wheel img endmetadatashow_logsèdisabled®skip_as_script«code_folded$416011b9-b4db-4c9e-9bfb-4dd105596733cell_id$416011b9-b4db-4c9e-9bfb-4dd105596733code!f2(z) = (z - 0.5 - 0.5*1im) / z^2metadatashow_logsèdisabled®skip_as_script«code_folded$80d4795c-c278-4844-a2cd-d375bc9330d1cell_id$80d4795c-c278-4844-a2cd-d375bc9330d1codemd"""## Pro Tip: The base function ⚠️ Notice that if we were to really plot the identify function, we would get a big black square because the density of the points in polar coordinates is not uniform. 👇 Try it out: Replace `custom_g` with the identity function"""metadatashow_logsèdisabled®skip_as_script«code_folded$cb82c7ac-1a9e-4d5b-ab4d-fb384f66b2eccell_id$cb82c7ac-1a9e-4d5b-ab4d-fb384f66b2eccodemd"And tadaaa 🎉"metadatashow_logsèdisabled®skip_as_script«code_folded$8e1c0295-d7e1-4bb0-8e50-62a6e6992011cell_id$8e1c0295-d7e1-4bb0-8e50-62a6e6992011code3function g3(r) return 0.5 + 0.5*(r - floor(r)) endmetadatashow_logsèdisabled®skip_as_script«code_folded$bd52785f-c057-4bc3-94ae-86e1dd773ba7cell_id$bd52785f-c057-4bc3-94ae-86e1dd773ba7codemd"""To finally create our art pieces, all we have to do is repeat the same thing we did so far for **all** the points in our 2D space. So instead of checking the values for each point one by one, we will do the following: 👉 Go over each point in the initial drawing one by one 🔵 👉 Apply the transformation from above 📏 👉 Put in the transformed color 🔴 in the place of the original one in the new drawing """metadatashow_logsèdisabled®skip_as_script«code_folded$b8fee414-320c-4268-95b4-90e53700e0b8cell_id$b8fee414-320c-4268-95b4-90e53700e0b8codebegin intensity_picker = @bind v PlutoUI.Slider(0:0.05:1, default=0.9, show_value=true) aside( md""" 👉 You can also change the intensity of the colors here: intensity = $intensity_picker """, v_offset= -380) endmetadatashow_logsèdisabled®skip_as_script«code_folded$48386799-c121-4358-b7fa-3988ddd3c0efcell_id$48386799-c121-4358-b7fa-3988ddd3c0efcodef1(z) = (z^3 - 1) / zmetadatashow_logsèdisabled®skip_as_script«code_folded$805529fa-a017-435d-949f-c718ed74b56fcell_id$805529fa-a017-435d-949f-c718ed74b56fcodefunction_choosermetadatashow_logsèdisabled®skip_as_script«code_folded$374d32d1-2d91-44a7-8624-7ca9f6af32f8cell_id$374d32d1-2d91-44a7-8624-7ca9f6af32f8codeUaside(md""" !!! info "Functions vs Brushes 🖌️🧑‍🎨" What we're doing here isn't very different than what an artist does: we're taking the initial canvas of colors and changing them by adding ➕ intensity, 🔄 rotating the colors, $\rightarrow$ moving some to the side. All of these are **transformations** that can be expressed using **only** numbers instead of brushes. So, when learning math, we're not *always* restricted to using only numbers but rather we can map these numbers to colors 🎨 and play around with those instead, how awesome is that ! 🤯 """, v_offset=100)metadatashow_logsèdisabled®skip_as_script«code_folded$993ec4e0-df4d-4720-b256-40b6f3e91f55cell_id$993ec4e0-df4d-4720-b256-40b6f3e91f55codef3(z) = 1/zmetadatashow_logsèdisabled®skip_as_script«code_folded$8a77f33c-34ee-45e4-8693-468cb5c8a345cell_id$8a77f33c-34ee-45e4-8693-468cb5c8a345codecustom_f(x) = x^2 #Replace thismetadatashow_logsèdisabled®skip_as_script«code_folded«notebook_id$f3fb5c1a-b8ac-11f1-8f4b-e367c4ebe32din_temp_dir¨metadatafrontmatterlicense_urlghttps://github.com/JuliaPluto/featured/blob/2a6a9664e5428b37abe4957c1dca0994f4a8b7fd/LICENSES/UnlicenseauthornameJuliaPlutourlhttps://github.com/JuliaPlutoimageOhttps://github.com/user-attachments/assets/0273c73f-f92f-4ceb-a03a-d8839983f591titleDomain Coloringdate2025-01-17tagsmathjuliaartcomplex numberstransformationsdescription4Let's use math as a paintbrush with complex numbers!licenseUnlicense