5 · Shape Traversal

Physical meaning of the three key VAE dimensions

Paper section: Results §3.3–3.4 · Notebooks: 2.4, 2.4b, 10, 11, 13

Overview

To interpret what each VAE dimension physically encodes, we decode silhouettes at 11 evenly-spaced points along each dimension axis, holding all others at their period-genre mean. We then measure five geometric descriptors at each step:

Descriptor	Formula	Captures
h/w ratio	height ÷ width	overall orientation
Bounding-box fill	pixel area ÷ bounding-box area	how compactly the tablet fills its extent
Aspect ratio of fill	fill-adjusted h/w	shape after compactness correction
Contour complexity	perimeter² ÷ (4π × area)	outline irregularity (1 = circle)
Solidity	convex-hull area ÷ pixel area	body unity vs. articulated projection

For each traversal we identify the primary descriptor — the one that changes most monotonically. The primary descriptor is the physical meaning of that dimension.

X7 — diplomatic-tradition axis

Primary descriptor (all genres): h/w ratio

import matplotlib.pyplot as plt, matplotlib.image as mpimg, os

fig_path = "../../paper/figures/fig_vae_traversal_X7_genre.pdf"
if os.path.exists(fig_path):
    print(f"Traversal filmstrip: {fig_path}")
    print("This figure shows decoded silhouettes at 11 steps along the X7 axis,")
    print("stratified by genre (Administrative / Literary / Legal).")
else:
    print("Figure not found. Run notebook 2.4b to generate.")

Traversal filmstrip: ../../paper/figures/fig_vae_traversal_X7_genre.pdf
This figure shows decoded silhouettes at 11 steps along the X7 axis,
stratified by genre (Administrative / Literary / Legal).

Figure 1

Figure 2: X7 traversal filmstrip — pure h/w ratio axis

X7 controls h/w ratio in isolation: as X7 decreases from +4 to −4, the tablet silhouette rotates smoothly from tall portrait to wide landscape, while its fill efficiency and outline complexity remain approximately constant. Critically, this behavior is genre-consistent: whether the baseline tablet is Administrative, Literary, or Legal, X7 traversal always produces the same portrait→landscape rotation.

Genre-consistency makes X7 the cleanest single-number summary of scribal orientation tradition. A period’s mean X7 value directly encodes whether its scribes worked in a portrait or landscape tradition, independent of genre composition.

Key period values:

Period	Mean X7	Tradition
Neo-Assyrian	+2.05	Extreme portrait
Ur III	+0.89	Portrait
Old Babylonian	+0.61	Mild portrait
Neo-Babylonian	−1.14	Landscape
Achaemenid	−2.48	Strong landscape

X2 — bounding-box fill efficiency (diachronic-drift axis)

Primary descriptor: fill efficiency (genre-inconsistent)

import matplotlib.pyplot as plt, os

fig_path = "../../paper/figures/fig_vae_traversal_X2_genre.pdf"
if os.path.exists(fig_path):
    print(f"Traversal filmstrip: {fig_path}")
    print("Administrative baseline: low X2 = articulated portrait, high X2 = compact landscape")
    print("Literary baseline: primary descriptor shifts — encodes genre-specific norms")
else:
    print("Figure not found. Run notebook 2.4b to generate.")

Traversal filmstrip: ../../paper/figures/fig_vae_traversal_X2_genre.pdf
Administrative baseline: low X2 = articulated portrait, high X2 = compact landscape
Literary baseline: primary descriptor shifts — encodes genre-specific norms

Figure 3

Figure 4: X2 traversal filmstrip — bounding-box fill efficiency

X2 captures a composite morphological feature:

• Low X2 (Ur III admin mean = −0.995): tall portrait silhouette with an articulated, irregular outline — fingers, projections, and surface unevenness reduce fill efficiency
• High X2 (Achaemenid admin mean = +0.652): compact landscape rectangle with a smooth, regular outline — high fill efficiency

X2 tracks time better than X7 (Spearman ρ = +0.655 vs. −0.430) precisely because it captures two simultaneous historical trends: 1. Portrait → landscape rotation (shared with X7) 2. Outline regularization (additional signal not in X7)

Genre-inconsistency (the primary descriptor shifts when you change the baseline genre) is evidence that X2 encodes genre-specific shape norms layered on top of the global diachronic trend — not a methodological failure.

X8 — body unity / silhouette cohesion

Primary descriptor: solidity (genre-inconsistent for Admin vs. others)

import matplotlib.pyplot as plt, os

fig_path = "../../paper/figures/fig_vae_traversal_X8_genre.pdf"
if os.path.exists(fig_path):
    print(f"Traversal filmstrip: {fig_path}")
    print("High X8: segmented outline with visible projections and divisions")
    print("Low X8: unified slab-like form, convex hull closely matches pixel area")
else:
    print("Figure not found. Run notebook 2.4b to generate.")

Traversal filmstrip: ../../paper/figures/fig_vae_traversal_X8_genre.pdf
High X8: segmented outline with visible projections and divisions
Low X8: unified slab-like form, convex hull closely matches pixel area

Figure 5

Figure 6: X8 traversal filmstrip — body unity axis

X8 controls silhouette cohesion: whether the tablet body appears as a single unified slab (low X8) or as an articulated form with projections and segments (high X8). This dimension tracks a separate historical trend — the progressive loss of the 3rd-millennium rounded, organic tablet form in favor of the 1st-millennium regularized, brick-like form.

Diachronic trend: Spearman ρ = −0.573 (p = 0.007), indicating X8 decreases (unifies) monotonically across time.

The discrimination vs. trend dissociation

The three dimensions reveal a structural finding:

import pandas as pd, matplotlib.pyplot as plt, numpy as np

dims = ['X2', 'X7', 'X8']
eta2 = [0.078, 0.435, 0.025]
abs_rho = [0.655, 0.430, 0.573]
labels = ['X2\n(fill efficiency)', 'X7\n(h/w ratio)', 'X8\n(body unity)']
colors = ['#b5622e', '#2c6e49', '#4a6fa5']

fig, axes = plt.subplots(1, 2, figsize=(10, 4))

# Left: eta2 comparison
axes[0].bar(labels, eta2, color=colors, alpha=0.85, edgecolor='white')
axes[0].set_ylabel('η² (period discrimination)', fontsize=9)
axes[0].set_title('Period discrimination power', fontsize=10)
axes[0].axhline(0.14, color='grey', ls='--', lw=0.8, alpha=0.6, label='Large effect')
axes[0].axhline(0.06, color='grey', ls=':', lw=0.8, alpha=0.6, label='Medium effect')
axes[0].legend(fontsize=7)

# Right: |rho| comparison
axes[1].bar(labels, abs_rho, color=colors, alpha=0.85, edgecolor='white')
axes[1].set_ylabel('|Spearman ρ| (diachronic trend)', fontsize=9)
axes[1].set_title('Diachronic trend strength', fontsize=10)
axes[1].axhline(0.5, color='grey', ls='--', lw=0.8, alpha=0.6, label='Strong trend')
axes[1].legend(fontsize=7)

plt.suptitle('X7: best discriminator; X2 & X8: best trend trackers', fontsize=11)
plt.tight_layout()
plt.show()

Figure 7: Three-panel summary: X7 discriminates traditions (η²=0.435) but shows no trend; X2 and X8 both trend across time (|ρ|>0.55) but discriminate less.

Dimension	η²		ρ
X7	0.435 (large)	0.430 (ns)	h/w ratio	Scribal tradition ID
X2	0.078 (medium)	0.655***	Fill efficiency	Diachronic tracking
X8	0.025 (small)	0.573**	Body unity	Secondary time signal

The discrimination vs. trend dissociation means that the dimension that best separates scribal traditions (X7) is not the dimension that best tracks time (X2), and vice versa. This is the latent-space analogue of distinguishing tradition from innovation: a scribal school can maintain its portrait orientation (high X7) while still moving toward more regularized outlines (increasing X2) over centuries.

Note

Next: Geographic Analysis → — whether the portrait-to-landscape shift is a Babylonian phenomenon or a pan-Mesopotamian one.