[EDIT: converted latex to unicode, because latex is not working]

I conjecture that tetration introduces numbers with fractional dimension Rᴿ = ²R.

For that purpose, I'm trying to extend Geometric algebra (Clifford algebra Cℓₙ,₀,₀), n ∈ N to fractional dimension, defining r-blades (of grade r ∈ R).

I attached a PDF explaining it. (it is a draft, not a finished text. And I'm not a mathematician).

In Clifford algebra, given two normal vectors (1-blades), their product should be anticommutative:

f · g = -g · f

Each vector can be decomposed into d fracvectors with fractional dimension D = 1/s (s ∈ N).

f = f₀ · f₁ · f₂ · ... · fₛ

g = g₀ · g₁ · g₂ · ... · gₖ

The problem I don't know how to fix is how to define the anticommutativity of fᵢ with gⱼ.

f · g = -g · f

but

f · g = f · g₀ · g₁ · g₂ · ... · gₖ

then f · g = -g · f should be preserved regardless of how many gᵢ there are (independently of the value of d, k, which determine the number of commutations of gᵢ with fⱼ).

Commuting

f · (g₀ · g₁) = -g · f

Should produce the same signs change than

f · (g₀ · g₁ · g₂) = -g · f

regardless of if g is decomposed in 2 or 3 fractional dimensions

I can't figure how to achieve that. Any idea?

I conjecture that tetration introduces numbers with fractional dimension Rᴿ = ²R.

For that purpose, I'm trying to extend Geometric algebra (Clifford algebra Cℓₙ,₀,₀), n ∈ N to fractional dimension, defining r-blades (of grade r ∈ R).

I attached a PDF explaining it. (it is a draft, not a finished text. And I'm not a mathematician).

In Clifford algebra, given two normal vectors (1-blades), their product should be anticommutative:

f · g = -g · f

Each vector can be decomposed into d fracvectors with fractional dimension D = 1/s (s ∈ N).

f = f₀ · f₁ · f₂ · ... · fₛ

g = g₀ · g₁ · g₂ · ... · gₖ

The problem I don't know how to fix is how to define the anticommutativity of fᵢ with gⱼ.

f · g = -g · f

but

f · g = f · g₀ · g₁ · g₂ · ... · gₖ

then f · g = -g · f should be preserved regardless of how many gᵢ there are (independently of the value of d, k, which determine the number of commutations of gᵢ with fⱼ).

Commuting

f · (g₀ · g₁) = -g · f

Should produce the same signs change than

f · (g₀ · g₁ · g₂) = -g · f

regardless of if g is decomposed in 2 or 3 fractional dimensions

I can't figure how to achieve that. Any idea?