Gemini Functions: A Geometric Dilogarithm Framework

• Gemini functions are a two-parameter family of self-inverse maps defined via logarithmic-coth formulations, unifying dilogarithm identities through area dissections.
• They feature integral representations with Li₂ and exhibit key symmetries such as reflection, inversion, Landen, and duplication formulas, ensuring their functional robustness.
• Special parameter choices yield classical identities and finite-area evaluations at algebraic points, offering concrete geometric and analytic applications.

Gemini functions are a family of self-inverse functions introduced by Alha as a geometric-analytic framework for deriving and organizing dilogarithm identities (Alha, 9 Sep 2025). The construction begins with the fundamental form

$\operatorname{gemini}_1(x):=\ln \coth(x/2),$

arising in hyperbolic-parallel-angle geometry, and extends to a two-parameter family $\operatorname{gemini}_a^b(x)$ with a “shape” parameter $a\ge -1$ and a “scale” parameter $b>0$. In the formulation given in the paper, these functions are characterized simultaneously by involutivity, integral representations involving $\operatorname{Li}_2$, and area dissections that yield a master five-term identity together with reflection, inversion, Landen, duplication, and fixed-point ladder formulas (Alha, 9 Sep 2025).

1. Fundamental form and two-parameter extension

The fundamental form is defined by

$\operatorname{gemini}_1(x):=\ln \coth(x/2).$

Equivalently, one writes

$$x\mapsto y:=\ln\left[\frac{1+e^{-x}}{1-e^{-x}}\right],$$

and the defining feature is that this map is involutive: swapping $x\leftrightarrow y$ reproduces the same formula (Alha, 9 Sep 2025).

The two-parameter extension introduces a “shape” parameter $a\ge -1$ and a “scale” parameter $b>0$, and is given by

$\operatorname{gemini}_a^b(x)$0

In the terminology of the paper, $\operatorname{gemini}_a^b(x)$1 controls the “steepness” of the curve, while $\operatorname{gemini}_a^b(x)$2 scales the horizontal and vertical axes: larger $\operatorname{gemini}_a^b(x)$3 stretches the graph while preserving its shape. When $\operatorname{gemini}_a^b(x)$4 and $\operatorname{gemini}_a^b(x)$5, one recovers the fundamental form. When $\operatorname{gemini}_a^b(x)$6, one obtains the degenerate form

$\operatorname{gemini}_a^b(x)$7

This parameterization is the basic organizational device for the theory. Rather than treating classical dilogarithm identities as isolated formulas, the paper places them inside a single family whose deformation parameters govern geometry, involution, and special-value phenomena (Alha, 9 Sep 2025).

2. Involution, kinds of symmetry, and functional structure

The central structural property is self-inversion. From the defining equation

$\operatorname{gemini}_a^b(x)$8

one obtains the equivalent relation

$\operatorname{gemini}_a^b(x)$9

Hence

$a\ge -1$0

for all $a\ge -1$1 in its domain (Alha, 9 Sep 2025).

The paper also states that one derives, in special regimes, further functional symmetries: in the degenerate case a reflection identity, in the fundamental case an inversion identity, and in related rotated configurations additional classical formulas. The involutive structure is therefore not merely a formal curiosity; it is the mechanism that makes area comparisons close algebraically into dilogarithm identities.

A useful consequence of this setup is that special choices of $a\ge -1$2, $a\ge -1$3, and distinguished $a\ge -1$4-values such as fixed points or medians produce families of identities with the same structural origin. This suggests a unification of several functional equations that are often presented separately.

3. Integral representation and finite-area property

For the fundamental form, the integral admits a two-term dilogarithm representation: $a\ge -1$5 The total area under the graph from $a\ge -1$6 to $a\ge -1$7 is finite, and the paper gives

$a\ge -1$8

(Alha, 9 Sep 2025).

In full generality, one has

$a\ge -1$9

The corresponding total area is stated as

$b>0$0

since $b>0$1 (Alha, 9 Sep 2025).

These formulas are the analytic core of the construction. Every subsequent identity is obtained by evaluating such integrals over geometrically selected intervals or regions. In this sense, the family is not defined only by a closed-form expression; it is also defined by an integral calculus in which $b>0$2 appears naturally as the area primitive.

4. Master five-term identity and classical corollaries

The main identity is obtained by dissecting the area under $b>0$3 into rectangular and “apex” parts. The resulting five-term formula is

$b>0$4

valid for $b>0$5 and $b>0$6. In the paper this is described as a generalization of the classical five-term relation of the dilogarithm (Alha, 9 Sep 2025).

Several standard identities arise as specializations or geometric rotations of this formula:

• Reflection (Euler) formula, degenerate case $b>0$7:

$b>0$8

• Inversion formula:

$b>0$9

• Landen’s identity:

$\operatorname{Li}_2$0

• Duplication formula:

$\operatorname{Li}_2$1

The significance of the master identity lies in its role as a generating relation. A single involutive family yields all four classical functional equations by elementary area comparisons, rather than by separate ad hoc manipulations.

5. Fixed points, ladders, and algebraic special values

A further source of identities comes from choosing the integration limit to be a fixed point $\operatorname{Li}_2$2 satisfying

$\operatorname{Li}_2$3

From this choice one obtains a three-term identity of the form

$\operatorname{Li}_2$4

(Alha, 9 Sep 2025).

The paper emphasizes that specializing $\operatorname{Li}_2$5 to algebraic constants yields closed-form values and finite ladders.

For the fundamental form $\operatorname{Li}_2$6, one recovers classical four-term identities and the values

$\operatorname{Li}_2$7

For the degenerate form $\operatorname{Li}_2$8, the fixed point is $\operatorname{Li}_2$9, and the same value

$\operatorname{gemini}_1(x):=\ln \coth(x/2).$0

is recovered from the area between $\operatorname{gemini}_1(x):=\ln \coth(x/2).$1 and $\operatorname{gemini}_1(x):=\ln \coth(x/2).$2 of $\operatorname{gemini}_1(x):=\ln \coth(x/2).$3.

The paper then lists several algebraic examples:

• For the golden ratio $\operatorname{gemini}_1(x):=\ln \coth(x/2).$4, setting $\operatorname{gemini}_1(x):=\ln \coth(x/2).$5 and choosing $\operatorname{gemini}_1(x):=\ln \coth(x/2).$6 yields

$\operatorname{gemini}_1(x):=\ln \coth(x/2).$7

• For the silver ratio $\operatorname{gemini}_1(x):=\ln \coth(x/2).$8, the fixed-point identity gives

$\operatorname{gemini}_1(x):=\ln \coth(x/2).$9

where

$$x\mapsto y:=\ln\left[\frac{1+e^{-x}}{1-e^{-x}}\right],$$0

• For the plastic constant $$x\mapsto y:=\ln\left[\frac{1+e^{-x}}{1-e^{-x}}\right],$$1, solving the “three-term” median condition or the “trinomial” ladder with $$x\mapsto y:=\ln\left[\frac{1+e^{-x}}{1-e^{-x}}\right],$$2 yields

$$x\mapsto y:=\ln\left[\frac{1+e^{-x}}{1-e^{-x}}\right],$$3

together with related two-term identities and ladders of length $$x\mapsto y:=\ln\left[\frac{1+e^{-x}}{1-e^{-x}}\right],$$4 for $$x\mapsto y:=\ln\left[\frac{1+e^{-x}}{1-e^{-x}}\right],$$5.

• Analogous choices involving the super-golden $$x\mapsto y:=\ln\left[\frac{1+e^{-x}}{1-e^{-x}}\right],$$6, the second Pisot $$x\mapsto y:=\ln\left[\frac{1+e^{-x}}{1-e^{-x}}\right],$$7, and roots of $$x\mapsto y:=\ln\left[\frac{1+e^{-x}}{1-e^{-x}}\right],$$8 generate six-term or seven-term ladders.

These examples show that the framework is not restricted to reproducing standard textbook identities; it also systematizes finite-argument evaluations at algebraic points.

6. Geometric interpretation, asymptotics, and unifying scope

The geometric interpretation is explicit. Every term $$x\mapsto y:=\ln\left[\frac{1+e^{-x}}{1-e^{-x}}\right],$$9 in a gemini identity corresponds to the signed area between certain $x\leftrightarrow y$0-limits under the graph of $x\leftrightarrow y$1. “Apex” areas, rectangular areas $x\leftrightarrow y$2, and the “middle-square” area $x\leftrightarrow y$3 all enter the algebra and enforce classical dilogarithm functional equations (Alha, 9 Sep 2025).

The median construction gives another canonical relation. If $x\leftrightarrow y$4 bisects the total area, then

$x\leftrightarrow y$5

In particular, the paper states that special $x\leftrightarrow y$6 arise for which $x\leftrightarrow y$7 is expressed in elementary constants.

The asymptotic regime $x\leftrightarrow y$8 is also described geometrically. As $x\leftrightarrow y$9, the graphs straighten between symmetric inflection points, the total-area ratio tends to $a\ge -1$0, matching two right-isosceles triangles, and the volume under $a\ge -1$1(dilog) tends to that under a circular cone, with volume-ratio $a\ge -1$2. In a related geometric direction, rotating $a\ge -1$3 about an axis yields a “geminoid” whose Gaussian curvature $a\ge -1$4 is computable in closed form; its generating meridian curve is involutive and of finite area $a\ge -1$5, and tangent-sweep (“Mamikon”) arguments recover that area.

Within the paper’s own summary, the unifying claim is precise: a single two-parameter family $a\ge -1$6 generates the reflection, inversion, Landen, and duplication formulas by elementary area comparisons; varying the shape parameter $a\ge -1$7 and selecting special points such as medians or fixed points produces known finite-argument dilogarithm ladders at algebraic values; and complex extensions with $a\ge -1$8 and $a\ge -1$9 are handled by the involutive geometry of rotated gemini and its two-parameter deformations (Alha, 9 Sep 2025). In this sense, gemini functions are presented not simply as isolated self-inverse maps, but as a geometric framework that subsumes and extends a substantial portion of the classical dilogarithm literature.