Create a new matrix from an existing matrix and then plot it

Question

I have some code will test an equation and finds values for (n,j,k,l) in which the expression is true and puts them in the matrix - here is the code.

m = Table[
If[(n)^2 + (j)^4 + (k)^3 + (l) == 0, {n, j, k, l}, Nothing], {n, 
 0, 10, 1}, {j, 0, 10, 1}, {k, -10, 0, 1}, {l, -10, 0, 1}] // 
   Flatten[#, 3] & // Select[Length@# > 0 &]; m // MatrixForm

Here is the output

So the columns of the matrix correspond to the values of (n,j,k,l). Now I want to create a new matrix, where I introduce a counter value that starts at say 0.01 and increments by 0.01 with each iteration, call it a++.

The new 3 column matrix will take the value of (n,j,a) and put it into the matrix as the first row, and also (while keeping the same a value) will also add (k,l,a) as the second row and then move the next row of the original matrix and so the same process, with the counter a incremented.

The code I made for this (although it does not work) is as follows

For[i = 1, i < 20, i++, m2[[i]] = {{m[[i, 1]]}, {m[[i, 2]]}, {i}}; 
m2[[i + 1]] = {{m[[i, 3]]}, {m[[i, 4]]}, {i}}]

Lastly, I want to use the 3 column matrix as (x,y,z) points to plot 3D point plot of surface in Mathematica

Answer 1

Something like this?

Join @@ MapIndexed[Partition[#1, 2] /. x_?VectorQ :> Join[x, 0.01 #2] &, m]
ListPlot3D[%, PlotStyle -> PointSize[0.02]]

{{0, 0, 0.01}, {0, 0, 0.01}, {0, 1, 0.02}, {-1, 0, 0.02}, {0, 1, 0.03}, {0, -1, 0.03}, {0, 2, 0.04}, {-2, -8, 0.04}, {1, 0, 0.05}, {-1, 0, 0.05}, {1, 0, 0.06}, {0, -1, 0.06}, {1, 1, 0.07}, {-1, -1, 0.07}, {1, 1, 0.08}, {0, -2, 0.08}, {1, 2, 0.09}, {-2, -9, 0.09}, {2, 0, 0.1}, {-1, -3, 0.1}, {2, 0, 0.11}, {0, -4, 0.11}, {2, 1, 0.12}, {-1, -4, 0.12}, {2, 1, 0.13}, {0, -5, 0.13}, {3, 0, 0.14}, {-2, -1, 0.14}, {3, 0, 0.15}, {-1, -8, 0.15}, {3, 0, 0.16}, {0, -9, 0.16}, {3, 1, 0.17}, {-2, -2, 0.17}, {3, 1, 0.18}, {-1, -9, 0.18}, {3, 1, 0.19}, {0, -10, 0.19}, {4, 0, 0.2}, {-2, -8, 0.2}, {4, 1, 0.21}, {-2, -9, 0.21}, {4, 2, 0.22}, {-3, -5, 0.22}, {6, 0, 0.23}, {-3, -9, 0.23}, {6, 1, 0.24}, {-3, -10, 0.24}, {7, 2, 0.25}, {-4, -1, 0.25}, {7, 3, 0.26}, {-5, -5, 0.26}, {8, 0, 0.27}, {-4, 0, 0.27}, {8, 1, 0.28}, {-4, -1, 0.28}}

Answer 2

An alternative way to generate m (seems to be faster than so-far-posted alternatives):

ClearAll[l, n, k, j]
m2 = Values @ Solve[{Total[{l, n, k, j}^Range[4]] == 0,
       -10 <= l <= 0 && -10 <= k <= 0 && 0 <= n <= 10 && 0 <= j <= 10},
      {n, j, k, l}, Integers]

m2 == m

True

new2 = ArrayReshape[MapIndexed[Riffle[#, .01 #2[[1]], {3, -1, 3}] &, m2],
   {2 Length @ m2, 3}]; 

Alternatively,

new2b = MapIndexed[## & @@ Partition[Riffle[#, .01 #2[[1]], {3, -1, 3}], 3] &] @ m2;
new2b == new2

True

ListPlot3D[new2]

Answer 3

If I understand correctly this is what you want. Not elegant though.

list = Flatten[ Outer[List, Range[0, 10], Range[0, 10], Range[-10, 0], Range[-10, 0]], 3];
m = Pick[list, #1^2 + #2^4 + #3^3 + #4 & @@@ list, 0]   

a = 0.01 Range[Length@m];
new = ArrayReshape[Transpose@Insert[Transpose@m, a, {{3}, {5}}], {2 Length@m, 3}]
ListPointPlot3D[new]