77 lines
2.1 KiB
Julia
77 lines
2.1 KiB
Julia
include("Engine.jl")
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using SparseArrays
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# this problem is from a sangaku by Irisawa Shintarō Hiroatsu. the article below
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# includes a nice translation of the problem statement, which was recorded in
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# Uchida Itsumi's book _Kokon sankan_ (_Mathematics, Past and Present_)
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#
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# "Japan's 'Wasan' Mathematical Tradition", by Abe Haruki
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# https://www.nippon.com/en/japan-topics/c12801/
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#
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# initialize the partial gram matrix
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J = Int64[]
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K = Int64[]
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values = BigFloat[]
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for s in 1:9
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# each sphere is represented by a spacelike vector
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push!(J, s)
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push!(K, s)
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push!(values, 1)
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# the circumscribing sphere is internally tangent to all of the other spheres
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if s > 1
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append!(J, [1, s])
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append!(K, [s, 1])
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append!(values, [1, 1])
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end
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if s > 3
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# each chain sphere is externally tangent to the "sun" and "moon" spheres
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for n in 2:3
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append!(J, [s, n])
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append!(K, [n, s])
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append!(values, [-1, -1])
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end
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# each chain sphere is externally tangent to the next chain sphere
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s_next = 4 + mod(s-3, 6)
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append!(J, [s, s_next])
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append!(K, [s_next, s])
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append!(values, [-1, -1])
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end
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end
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gram = sparse(J, K, values)
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# make an initial guess
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guess = hcat(
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Engine.sphere(BigFloat[0, 0, 0], BigFloat(15)),
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Engine.sphere(BigFloat[0, 0, -9], BigFloat(5)),
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Engine.sphere(BigFloat[0, 0, 11], BigFloat(3)),
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(
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Engine.sphere(9*BigFloat[cos(k*π/3), sin(k*π/3), 0], BigFloat(2.5))
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for k in 1:6
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)...
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)
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frozen = [CartesianIndex(4, k) for k in 1:4]
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# complete the gram matrix using Newton's method with backtracking
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L, success, history = Engine.realize_gram(gram, guess, frozen)
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completed_gram = L'*Engine.Q*L
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println("Completed Gram matrix:\n")
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display(completed_gram)
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if success
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println("\nTarget accuracy achieved!")
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else
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println("\nFailed to reach target accuracy")
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end
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println("Steps: ", size(history.scaled_loss, 1))
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println("Loss: ", history.scaled_loss[end], "\n")
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if success
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println("Chain diameters:")
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println(" ", 1 / L[4,4], " sun (given)")
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for k in 5:9
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println(" ", 1 / L[4,k], " sun")
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end
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end |