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]
Z=(x^3+y^3)^[1/3]
'z' Values coloured from the minimum (blue) to the maximum (red)
V=x^3+y^3-z^3
These are coloured based on how far V is from 0.
Zero is white, and the largest V % 9 encountered is blue. Interpolation is linear.
V and S will define a range that includes 0.
DIV 2
DIV 3
DIV 4
Subvalue=x^3+y^3-(z+1)^3
These are coloured based on how far subvalue is from 0.
Zero is white, and the most negative value encountered is red. Interpolation is linear.
If the calculations are correct, the subvalue and value will define a range that includes 0.
Intercept
Offset to the 'z' value where the line joining (z, value) to (z+1, subvalue) intercepts 0, an approximation to the cube root
of x^3+y^3.
(x^3+y^3+z^3) modulo 9.
There are nine possible values but 4 and 5 never occur. The other 7 are coloured using the colours of the rainbow:
0:"red", 1:"orange", 2:"yellow", 3:"green", 6:"blue", 7:"indigo", 8:"violet".
The corresponding frequencies (out of 27) are 7,6,3,1,0,0,1,3,6.
Close to 1/2
Highlights cells where the intercept is close to 0.5. Multiplying the values by two will generate a cell where the value is
closer to 0.
Close to 1/3
Highlights cells where the intercept is close to a multiple of 1/3. Multiplying the values by three will generate a cell where the value is
closer to 0.
Close to 1/4
Highlights cells where the intercept is close to a multiple of 1/4. Multiplying the values by four will generate a cell where the value is
closer to 0.
Close to 1/5
Highlights cells where the intercept is close to a multiple of 1/5. Multiplying the values by five will generate a cell where the value is
closer to 0.
Contours
If you reformulate the equation as x^3+y^3-(x+n)^3, where x>y, the the 'n' values define contours.