Author Archives: perfectionatic

Visualizing the Inscribed Circle and Square Puzzle

By: perfectionatic

Re-posted from: http://perfectionatic.org/?p=517

Recently, I watched a cool mind your decsions video on an inscribed circle and rectangle puzzle. In the video they showed a diagram that was not scale. I wanted to get a sense of how these differently shaped areas will match.

There was a cool ratio between the outer and inner circle radii that is expressed as

\frac{R}{r}=\sqrt{\frac{\pi-2}{4-\pi}} .

I used Compose.jl to rapidly do that.

using Compose
set_default_graphic_size(20cm, 20cm)
ϕ=sqrt((pi -2)/(4-pi))
R=10
r=R/ϕ
ctx=context(units=UnitBox(-10, -10, 20, 20))
composition = compose(ctx,
    (ctx, rectangle(-r/√2,-r/√2,r*2,r*2),fill("white")),
    (ctx,circle(0,0,r),fill("blue")),
    (ctx,circle(0,0,R),fill("white")),
    (ctx,rectangle(-10,-10,20,20),fill("red")))
composition |> SVG("inscribed.svg")

Solving the code lock riddle with Julia

By: perfectionatic

Re-posted from: http://perfectionatic.org/?p=494

I came across a neat math puzzle involving counting the number of unique combinations in a hypothetical lock where digit order does not count. Before you continue, please watch at least the first minute of following video:

The rest of the video describes two related approaches for carrying out the counting. Often when I run into complex counting problems, I like to do a sanity check using brute force computation to make sure I have not missed anything. Julia is fantastic choice for doing such computation. It has C like speed, and with an expressiveness that rivals many other high level languages.

Without further ado, here is the Julia code I used to verify my solution the problem.

  1. function unique_combs(n=4)
  2.     pat_lookup=Dict{String,Bool}()
  3.     for i=0:10^n-1
  4.         d=digits(i,10,n) # The digits on an integer in an array with padding
  5.         ds=d |> sort |> join # putting the digits in a string after sorting
  6.         get(pat_lookup,ds,false) || (pat_lookup[ds]=true)
  7.     end
  8.     println("The number of unique digits is $(length(pat_lookup))")
  9. end

In line 2 we create a dictionary that we will be using to check if the number fits a previously seen pattern. The loop starting in line 3, examines all possible ordered combinations. The digits function in line 4 takes any integer and generate an array of its constituent digits. We generate the unique digit string in line 5 using pipes, by first sorting the integer array of digits and then combining them in a string. In line 6 we check if the pattern of digits was seen before and make use of quick short short-circuit evaluation to avoid an if-then statement.

Julia calling C: A more minimal example

By: perfectionatic

Re-posted from: http://perfectionatic.org/?p=470

Earlier I presented a minimal example of Julia calling C. It mimics how one would go about writing C code, wrapping it a library and then calling it from Julia. Today I came across and even more minimal way of doing that while reading an excellent blog on Julia’s syntactic loop fusion. Associated with the blog was notebook that explores the matter further.

Basically, you an write you C in a string and pass it directly to the compiler. It goes something like

C_code= """
       double mean(double a, double b) {
         return (a+b) / 2;
       }
       """
const Clib=tempname()
open(`gcc -fPIC -O3 -xc -shared -o $(Clib * "." * Libdl.dlext) -`, "w") do f
     print(f, C_code)
end

The tempname function generate a unique temporary file path. On my Linux system Clib will be string like "/tmp/juliaivzRkT". That path is used to generate a library name "/tmp/juliaivzRkT.so" which will then used in the ccall:

julia> x=ccall((:mean,Clib),Float64,(Float64,Float64),2.0,5.0)
3.5

This approach would be be recommended if are writing anything sophisticated in C. However, it fun to experiment with for short bits of C code that you might like to call from Julia. Saves you the hassle of creating a Makefile, compiling, etc…