Cubes Sums

This page surfs the surface defined by the function Zv(x,y), which for a given position (x,y) returns the largest integer value of 'z' where x^3+y^3-z^3 is positive. There is a matching function Zs(x,y), which returns the smallest integer value of 'z' where x^3+y^3-z^3 is negative. We can then define two values, V and S as V(x,y)=x^3+y^3-Zv(x,y)^3 and S(x,y)=x^3+y^3-Zs(x,y)^3. We can convert S to a positive value negating the equation, −S(x,y)=Zs(x,y)^3-x^3-y^3. We are searching for small positive values of V and −S.

These surfaces are flat everywhere except at a set of edges where Zv and Zs change by 1, so you will have a surface that looks like it's been built from stacked cubes. At all points other than at the steps Zs=Zv+1. And the smooth surface x^3+y^3-z^3=0 lies between the two. They never intersect at integer value of x and y unless one or both of x and y are zero. We are only interested in the integer values of x and y.

This page allows you to explore this surface, with four views, one that shows an 11x11 grid of values and 3 that show different graphical views at different resolution . The underlying maths uses a variable length integer that allows you to explore arbitrarily large values. Also, to generate the images in a reasonable time (we are using Javascript in a browser, and the largest image contains a million points) we us a surfing algorithm. Once we have a value for Zv we can calculate the adjacent ones by adding the deltas, a bit like derivatives, but based on increments of one, rather than infinitesimals.

Value view

X Y Grid: Click actions:

Show:

grid

11x11 grid for x

11x11 Visualisation

Draw:


[hover mouse over grid to see values]

101x101 Visualisation

Draw:


[hover mouse over grid to see values]

1001x1001 Visualisation

Draw:


[hover mouse over grid to see values]

 

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(c) John Whitehouse 2021