Appendix D: Notation and Conventions
This appendix collects the notation used throughout the book. Conventions are consistent with the Geometric Series.
Manifolds and Spaces
| Symbol |
Meaning |
First Appearance |
| $\mathcal{P}$ |
Political preference manifold (6-dimensional) |
Ch. 4 |
| $d$ |
Dimensionality of $\mathcal{P}$ ($d = 6$) |
Ch. 4 |
| $d_1, \ldots, d_6$ |
The six manifold dimensions: economic, social, environmental, foreign policy, trust, identity |
Ch. 1 |
| $g_{ij}$ |
Riemannian metric tensor on $\mathcal{P}$ |
Ch. 4 |
| $\Sigma$ |
Covariance matrix of voter distribution on $\mathcal{P}$ ($6 \times 6$) |
Ch. 4 |
| $\Sigma_{ij}$ |
Covariance between dimensions $d_i$ and $d_j$ |
Ch. 4 |
| $\lambda_1, \ldots, \lambda_6$ |
Eigenvalues of $\Sigma$ (in decreasing order) |
Ch. 4 |
| $d_\mathcal{P}(v_1, v_2)$ |
Geodesic distance between $v_1$ and $v_2$ on $\mathcal{P}$ |
Ch. 4 |
| $d_M(v_1, v_2)$ |
Mahalanobis distance (manifold distance approximation) |
Ch. 4 |
| $T\mathcal{P}$ |
Tangent bundle of $\mathcal{P}$ |
Ch. 6 |
| $\mathcal{O}$ |
Outcome space of a voting function |
Ch. 5 |
| $W$ |
Overton window (connected bounded subset of $\mathcal{P}$) |
Ch. 10 |
| $\partial W$ |
Boundary of the Overton window |
Ch. 10 |
Voters and Representatives
| Symbol |
Meaning |
First Appearance |
| $V$ |
Set of voters |
Ch. 5 |
| $n$ |
Number of voters |
Ch. 5 |
| $v, v_i$ |
Individual voter (or voter’s manifold position) |
Ch. 4 |
| $\phi: V \to \mathcal{P}$ |
Preference configuration (assigns manifold position to each voter) |
Ch. 5 |
| $\mu$ |
Mean of voter distribution on $\mathcal{P}$ |
Ch. 4 |
| $\bar{v}$ |
Frechet mean of the voter distribution |
Ch. 4 |
| $r(v)$ |
Representative of voter $v$ under electoral system $E$ |
Ch. 15 |
| $r^*(v)$ |
Manifold-optimal representative for voter $v$ |
Ch. 15 |
| $x_C, x_{C_i}$ |
Candidate’s position on $\mathcal{P}$ |
Ch. 5 |
Voting and Elections
| Symbol |
Meaning |
First Appearance |
| $\mathcal{V}$ |
Voting function ($\mathcal{P}^n \to \mathcal{O}$) |
Ch. 5 |
| $E$ |
Electoral system |
Ch. 15 |
| $C_1, \ldots, C_m$ |
Candidates |
Ch. 5 |
| $m$ |
Number of candidates |
Ch. 5 |
| $k$ |
Number of districts |
Ch. 9 |
| $D$ |
Districting function ($V \to \{1, \ldots, k\}$) |
Ch. 9 |
| $D^{-1}(j)$ |
Voters assigned to district $j$ |
Ch. 9 |
Indices and Measures
| Symbol |
Meaning |
First Appearance |
| $BI(S, E)$ |
Political Bond Index for population $S$ under system $E$ |
Ch. 15 |
| $BI_i(S, E)$ |
Dimension-specific Bond Index on dimension $d_i$ |
Ch. 15 |
| $GI(D)$ |
Gerrymandering Index for districting $D$ |
Ch. 9 |
| $GI_{d_i}(D)$ |
Dimension-specific gerrymandering index |
Ch. 9 |
| $R$ |
Information preservation ratio |
Ch. 5, 13 |
| Symbol |
Meaning |
First Appearance |
| $\mathbf{F}_C$ |
Campaign heuristic field (vector field on $\mathcal{P}$) |
Ch. 6 |
| $\mathbf{e}$ |
Projection axis (unit vector in $\mathbb{R}^d$) |
Ch. 6 |
| $\mathbf{e}^*$ |
Optimal campaign projection axis |
Ch. 6 |
| $\mathbf{h}$ |
Political heuristic field (superposition of all sources) |
Ch. 6 |
| $\alpha_k(v)$ |
Influence weight of information source $k$ at voter $v$ |
Ch. 6 |
| $\beta_O$ |
Overton window boundary penalty coefficient |
Ch. 10 |
Symmetry and Gauge Theory
| Symbol |
Meaning |
First Appearance |
| $G_D$ |
Democratic gauge group |
Ch. 5 |
| $\tau_A$ |
Voter anonymity transformation |
Ch. 5 |
| $\tau_N$ |
Option neutrality transformation |
Ch. 5 |
| $\tau_R$ |
Re-description invariance transformation |
Ch. 5 |
| $D_4$ |
Dihedral group of order 8 (Hohfeldian symmetry) |
Ch. 17 |
Coalition Analysis
| Symbol |
Meaning |
First Appearance |
| $C$ |
Coalition (subset of voters or agents) |
Ch. 12 |
| $A_C$ |
Agreement submanifold of coalition $C$ |
Ch. 12 |
| $F(C)$ |
Fragility index of coalition $C$ |
Ch. 12 |
| $\epsilon$ |
Agreement tolerance threshold |
Ch. 12 |
Conventions
- Indices: Latin indices $i, j, k$ range over manifold dimensions (1 to 6). Subscripts on voter variables ($v_1, v_2$) index individuals.
- Vectors: Boldface ($\mathbf{e}, \mathbf{h}, \mathbf{F}$) for vectors and vector fields. Italic ($v, p$) for points on the manifold.
- Tensors: Component notation $g_{ij}$, $\Sigma_{ij}$ with Einstein summation convention where applicable.
- Expectations: $\mathbb{E}[\cdot]$ for expectations over the voter population; $\mathbb{E}_v[\cdot]$ when the variable of integration is specified.
- Series references: GR = Geometric Reasoning. GE = Geometric Ethics. GComm = Geometric Communication. GEcon = Geometric Economics. GMed = Geometric Medicine. GL = Geometric Law.