Seven Lines of Julia (examples sought)

I love watching particles dance! Awesome Leandro!

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Infinizoom! (Background: Simulates a Brownian motion and uses the Brownian self-similarity https://en.wikipedia.org/wiki/Wiener_process#Self-similarity combined with simple data-augmentation/Brownian bridge sampling to zoom in infinitely.)

using Makie
dt = 0.0005; ϵ = 0.1; t = collect(-1.0:dt:1.0); n = length(t)
x = sqrt(dt)*cumsum(randn(n)); x .-= x[end÷2]
T, X = Node(t), Node(x)
p = lines(T, X); display(p)
xlims!(-1.0, 1.0); ylims!(-1.5,1.5)
while true
    t .*= (1 + ϵ)
    x .*= sqrt(1 + ϵ)
    if t[end] > 2.0
        t .= collect(-1.0:dt:1.0) 
        x[1:2:end] = x[n÷4+1:3n÷4+1]
        x[2:2:end] = (x[1:2:end-2]+x[3:2:end])/2 + sqrt(dt)/2*randn(n÷2)
        x[end÷2+1] = 0
    end
    T[] = t; X[] = x; sleep(0.02)
end

18 Likes

@FedericoStra, found interesting to perform a zoom-in loop and recompute your fractal and to use animated gif technique shown by @leandromartinez98 to create:
mandelbrot_zooms

xc, yc = -0.55, 0.61; 
x0, x1 = xc - 2, xc + 2; y0, y1 = yc - 2, yc + 2;
anim = @animate for t in 1:50
    x, y = range(x0, x1; length=1000), range(y0, y1; length=1000)
    heatmap(x, y, -log.(mandelbrot.(x' .+ y .* im));aspect_ratio=1,border=:none,legend=:none);
    x0, x1 = (15x0 + x1)/16, (15x1 + x0)/16;  y0, y1 = (15y0 + y1)/16, (15y1 + y0)/16 
end
gif(anim,"mandelbrot_zooms.gif",fps=10)
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for reference, what is the extended code that gives your example (a) parallelized (b) with the actual L-J potential © both

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Nice! I was thinking of doing something similar, but I couldn’t find the time, nor fit it in less than 7 lines (you cheated! :stuck_out_tongue:).

I think that, to make it better, the number of iterations in mandelbrot has to increase the more one zooms in, otherwise the boundary of the fractal becomes too imprecise. Here is a complete version that does so:

using Plots
exprange(start, stop, len) = exp.(range(log(start), log(stop), length=len))
function mandelbrot(z; lim=75) w = z
    for n = 1:lim;  abs2(w) < 4 ? w = w^2 + z : return n end
    lim + 1
end
x₀, y₀ = -0.5626805, 0.6422555
anim = @animate for (r, l) in zip(exprange(2, 1.35e-6, 120), exprange(100, 2500, 120))
    x = range(x₀-r, x₀+r; length=600); y = range(y₀-r, y₀+r; length=600);
    heatmap(x, y, -log.(log.(mandelbrot.(x' .+ y .* im; lim=round(l))));
        legend=:none, border=:none, ticks=:none, size=(600,600), ratio=1)
end
g = gif(anim; fps=12)

Apparently I cannot attach the animated GIF file (which you can find here), so I include only the last frame:

8 Likes

Calculating a cross-correlation in any number of dimensions:

for I in CartesianIndices(img)
    for J in CartesianIndices(kernel)    # kernel with centered indices
        if I+J in CartesianIndices(img)
            filtered[I] += img[I+J] * kernel[J]
        end
    end
end

(Inspired by https://julialang.org/blog/2016/02/iteration/ and https://julialang.org/blog/2017/04/offset-arrays/.)

The code in action:

using OffsetArrays
using Makie

# A 3D "image" of random values (either 0 or 1)
img = rand([0.0, 1.0], 20, 60, 40)

# Plot
s1 = volume(img, algorithm=:iso, resolution=(800, 500))

# Function to generate an N-dimensional kernel of uniform values that sum to 1
# The center of the hypercube will have indices (0, 0, ..., 0)
uniform_kernel(n, dim) = OffsetArray(fill(1/n^dim, fill(n, dim)...), fill(-n÷2-1, dim)...)

# 5x5x5 array of uniform values that sum to 1
kernel = uniform_kernel(5, 3) # 3 dimensions

filtered = zero(img)

for I in CartesianIndices(img)
    for J in CartesianIndices(kernel)
        if I+J in CartesianIndices(img)
            filtered[I] += img[I+J] * kernel[J]
        end
    end
end

s2 = volume(filtered, algorithm=:iso, resolution=(800, 500))

Original image:
image

Filtered image: filtered

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What about a simulation with balls bouncing against each other?
Or a simulation of moving people getting infected with a virus when they meet other people.

I have one of those, but not in a compact code.

From here: https://github.com/m3g/CKP

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@LaurentPlagne posted this beautiful example:

See Plot a circle with a given radius with Plots.jl. I don’t know if he has published the code for it though.

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] activate .
instantiate

This is some of my favorite Julia code.

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Thank you for the comment :wink:

Unfortunately I don’t think that it can fit the 7 lines example list (>1000 SLOC)…

We use this example to show how to structure a not-too-small project (PkgSkeleton, types, modules, sub-modules, tests, export, dependencies, multiple dispatch, profile, //isme …) in our Julia training lectures.

Although I enjoy reading the examples in this thread, I would second @Tamas_Papp saying that Julia’s power is more strongly demonstrated through slightly larger examples.

We must take some time to refactor the package from its training material form and publish it : now I have an extra motivation to see if a Julia wizard could compact it significantly :smiley:

6 Likes

Side note: Love the color scheme. What are you using for it?

I think seven lines of code is excellent. It showcases what Julia can do, and hopefully shows how Jukia can express scientific notation or equations in a compact way.

However one small pleading from me. Let’s now go down the road of Obfuscated Perl. Back in the day when Perl was a dominant language for Web programming, and many other things, there were a lot of smart people involved. Still are probably! There were regular ‘Obfuscated Perl’ competitions.
Also the whitespace as code stuff (where you translate code into …err… meaningful whitespace ) etc.
I know I am being like the Grinch but IMHO these did the language no good

Don’t get me wrong - there is nothing bad about using a language feature but please, please explain what it does so other people can learn.

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I recently started using SatelliteToolbox.jl, so here is my submission. This code calculates the location (latitude, longitude, altitude) of the ISS “right now”. The readability suffers a bit trying to stick to seven, but it’s not too bad. This is based off of the example here (see the link for full details).

using Dates, SatelliteToolbox  
# Download data for ISS
download("https://celestrak.com/NORAD/elements/stations.txt", joinpath(pwd(), "space_stations.txt")) 
# Initialize the orbit propagator
orbp = init_orbit_propagator(Val(:sgp4), read_tle("space_stations.txt")[1])
# Get the current time (Julian date)
rightnow = DatetoJD(now())
# Propogate the orbit to "right now"
o, r_teme, v_teme = propagate_to_epoch!(orbp, rightnow)
# Get the position (radians, radians, meters)
lat, lon, h = ECEFtoGeodetic(rECItoECEF(TEME(), ITRF(), rightnow, get_iers_eop())*r_teme)
# Nice print out after conversions
println("Current location of the ISS: $(rad2deg(lat))°  $(rad2deg(lon))°  $(h/1000) km")
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@mihalybaci Is Santa tracking data available on 24 December?

:laughing:

Not sure, I poked around here a bit, and didn’t see anything for Santa. But this site gets its data from NORAD, and they’re the ones that do the Santa tracking, so it could be hidden in a file somewhere.

Möbius discovered the one-sided non-orientable strip in 1858 while serving as the chair of astronomy at the University of Leipzig. The concept inspired mathematicians in the development of the field of topology as well as artists like M.C. Escher.
Fifteen lines of Julia were required to compute and plot the Mobius strip and a moving normal vector, which changes color after each revolution. This exceeds the limit but someone might be able to compress it further.
(PS: edited code for better readability. Without sleep, the last for loop could be written as a comprehension too)

using Makie, ForwardDiff, LinearAlgebra
const FDd = ForwardDiff.derivative

M(u,v) = ((r + v/2*cos(u/2))*cos(u), (r + v/2*cos(u/2))*sin(u), v/2*sin(u/2))
∂M∂u(u,v) = [FDd(u->M(u,v)[1],u), FDd(u->M(u,v)[2],u), FDd(u->M(u,v)[3],u)]
∂M∂v(u,v) = [FDd(v->M(u,v)[1],v), FDd(v->M(u,v)[2],v), FDd(v->M(u,v)[3],v)]

r = 1.0; w = 0.5; m = 90; clr = (:cyan,:red,:cyan,:red,:cyan)
u = range(0, 8π, length = 4m); v = range(-w, w, length = 3)

N = [ -cross(∂M∂u(ui, 0.0), ∂M∂v(ui, 0.0))[j] for ui in u, j in 1:3 ]
N = N ./ sqrt.(sum(abs2,N,dims=2));  N = vec.(Point3f0.(N[:,1],N[:,2],N[:,3]))
xs, ys, zs = [[p[i] for p in M.(u, v')] for i in 1:3]
P0 = vec.(Point3f0.(xs[:,2], ys[:,2], zs[:,2]))

surface(xs[1:m+1,:], ys[1:m+1,:], zs[1:m+1,:], backgroundcolor=:black)
for i in 1:4m
    sleep(0.01); k = 1 + i÷(m+0.5);
    arrows!([P0[i]],[N[i]],lengthscale=0.3,arrowsize=0.05,arrowcolor=clr[k],linecolor=clr[k])
end

24 Likes

Instead of compressing it further - I’d be more interested to see it broken out to be more readable :). Awesome plot though!

5 Likes

For what it is worth, Makie’s ˋbandˋ function actually makes 2d bands in 3d space… comes in handy for your Möbius strips:

t = 0:0.1:2pi+0.1
lower = [Point3f0(.2cos(3s),0.0, 0.0) .+ (1 + 0.2sin(3s))*Point3f0(0, cos(s), sin(s)) for s in t]
upper = [Point3f0(-.2cos(3s),0.0, 0.0) .+ (1 - 0.2sin(3s)).*Point3f0(0, cos(s), sin(s)) for s in t]
band(lower, upper, color = [1.0:length(t)÷2; length(t)÷2:-1:1.0; length(t)÷2:-1:1.0; 1.0:length(t)÷2])

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Thanks. As a curiosity, plotting the normal around that new strip seems to indicate that it is orientable?

6 Likes