Definition of K_α(x)

(Inspired from Conway´s Base-13 function - see)

K_α(x) ∈ ℝ; x ∈ ℝ; α ∈ ℝ; 0 < α < π/2

Imagine x written as infinit Base-12 number

Def: e ∈ { 0, 1, 2, ... 8, 9, A, B } (Base-12 digit)

Def: d ∈ { 0, 1, 2, ... 8, 9 } (decimal digit)

x in Base-12 has the Form: ±e⁺.e^*

From now on ignore the point '.' and the sign in front - only look at the pure digits e⁺

We will generate n ∈ ℕ from e⁺ and we are lookingh for the very last A or B digit.

row_α,n(x) = x tanα + n/cosα

col_α,n(x) = x cotα + n/sinα

Definition of K₀(x) and K_π/2(x)

K_π/2(x) = K₀(x)

First we define K(x, ε); ε ∈ ℝ; 0 < ε ⋘ 1

Def: C(x) ist Conway´s Base-13 function.

Take x as Base-12 as before, ignore point and sign and get e⁺

if x - ⎣x⎦ < ε :	K(x, ε) = C(x)
if e+ = e*Ad*Ad* :	K(x, ε) = n	(n = decimal number from d* inbetween A and A)
if e+ = e*Bd*Ad* :	K(x, ε) = -n	(n = decimal number from d* inbetween B and A)
if e+ = e*Ad*Bd* :	K(x, ε) = n	(n = decimal number from d* inbetween A and B)
if e+ = e*Bd*Bd* :	K(x, ε) = -n	(n = decimal number from d* inbetween B and B)
else :	K(x, ε) = 0

K₀(x) = K(x, ε) for ε → 0