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Sep 30, 2021, 7:48:12 AM9/30/21

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Mandelbrot's negative dimensions and Lapidus' complex dimensions

have extended our concept of dimensions.

The isomers of the Moran dimension:

s=Log[2]/Log[7^(1/4)]=Log[4]/Log[Sqrt[7]]=Log[16]/Log[7]

suggest an extended Moran self-similarty form:

n^p/(m^p)^s=1

to give fractal rational integer Moran dimensions:

s[n_, m_, p_, q_] = N[Log[n^p]/Log[m^q]]

m<=2: m ratio

n>=m: n number of transforms

p and q integer or rational powers as q->1/q

When plotted this distribution of dimension has 2053 sequence values

between s=0 and s=25, for n,m<=10.

https://www.wolframcloud.com/obj/rlbagulatftn/Published/rational_Moran_Dimensional_sequence_both_q.nb

https://www.pinterest.com/pin/293648838212628366/

have extended our concept of dimensions.

The isomers of the Moran dimension:

s=Log[2]/Log[7^(1/4)]=Log[4]/Log[Sqrt[7]]=Log[16]/Log[7]

suggest an extended Moran self-similarty form:

n^p/(m^p)^s=1

to give fractal rational integer Moran dimensions:

s[n_, m_, p_, q_] = N[Log[n^p]/Log[m^q]]

m<=2: m ratio

n>=m: n number of transforms

p and q integer or rational powers as q->1/q

When plotted this distribution of dimension has 2053 sequence values

between s=0 and s=25, for n,m<=10.

https://www.wolframcloud.com/obj/rlbagulatftn/Published/rational_Moran_Dimensional_sequence_both_q.nb

https://www.pinterest.com/pin/293648838212628366/

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