### Generalizing the Golden Ratio Sequence to Higher Dimensions

Posted:

**Fri Feb 20, 2015 10:52 pm**I am trying to figure out a way to generalize the golden ratio sequence to higher dimensions in a way that is extensible(does not make any assumption about the amount of samples that are going to taken), as well as scaling well to 6 or more dimensions.

The golden ratio sequence is desirable because it uses the irrationality of golden ratio to avoid the structured noise and moire artifacts of other sequences, and it is the cheapest low discrepancy sequence known, requiring only an addition and check for overflow.

update: I have tried doing 2 steps for the second, 1 for first, the result is that the diagonal resulting from using the same values for both x and y just repeats twice instead of once.

The original paper https://www.graphics.rwth-aachen.de/pub ... /2/jgt.pdf uses a permutation like the faure sequence to generalize sequence to 2 dimensions,

which assumes that i know how many samples are going to be taken, which I do not want to know.

A later paper http://www.researchgate.net/profile/Col ... 000000.pdf generalizes the sequence using a Hilbert Curve,

which I am not so confident about the performance in 32 bit precision, especially 6 or more dimensions(tell me if I am wrong) or the quality of it mapped to a sphere compared to the original.

One idea for the unit square i have is to take inspiration from the Halton sequence

What i would prefer is one generalization for the unit square and another one specifically for the unit sphere.

The golden ratio sequence is desirable because it uses the irrationality of golden ratio to avoid the structured noise and moire artifacts of other sequences, and it is the cheapest low discrepancy sequence known, requiring only an addition and check for overflow.

update: I have tried doing 2 steps for the second, 1 for first, the result is that the diagonal resulting from using the same values for both x and y just repeats twice instead of once.

The original paper https://www.graphics.rwth-aachen.de/pub ... /2/jgt.pdf uses a permutation like the faure sequence to generalize sequence to 2 dimensions,

which assumes that i know how many samples are going to be taken, which I do not want to know.

A later paper http://www.researchgate.net/profile/Col ... 000000.pdf generalizes the sequence using a Hilbert Curve,

which I am not so confident about the performance in 32 bit precision, especially 6 or more dimensions(tell me if I am wrong) or the quality of it mapped to a sphere compared to the original.

One idea for the unit square i have is to take inspiration from the Halton sequence

What i would prefer is one generalization for the unit square and another one specifically for the unit sphere.