Polyhedral Family·Inventor: Hajime Narukawa·Year: 1999

AuthaGraph Projection

"Equal-area tetrahedral projection tessellating seamlessly in 2D space without dead ends"

Created by Japanese architect Hajime Narukawa, AuthaGraph divides the spherical earth into 96 regions, projects them onto an inflated tetrahedron, and unfolds into an approximate equal-area rectangle that can tile infinitely.

Forward Mathematical Formulation

X-COORDINATE:

Spherical tessellation onto 96 spherical triangles + tetrahedral flattening

Y-COORDINATE:

Equal-area transformation preserving continental aspect ratios

DIFFERENTIAL METRIC:

Tessellation without boundary discontinuity in continuous 2D planar arrays

Cartographic Application & Critique

Primary Applications: Japanese high school official geography textbooks (since 2011), Good Design Award Grand Prize (2016).

Critical Analysis: Eliminates the arbitrary Eurocentric center of world maps; can be framed with Antarctica at center, Pacific at center, or Atlantic at center simply by panning the endless tessellation.

TISSOT INDICATRIX (Lat: 60°N)Area: 1.02×

MERIDIAN (h)

1.13

PARALLEL (k)

0.90

ANGULAR (2θ')

13.2°

Tissot Profile Across Latitudes

LATITUDE	SCALE h	SCALE k	AREA hk	2θ' DEFORM
0°N	1.03	0.97	1.00×	6.2°
30°N	1.04	0.96	1.00×	8.1°
60°N	1.06	0.94	1.00×	12.0°
80°N	1.10	0.91	1.00×	18.4°

Interdisciplinary Geometric Links

MorphologiumDERIVES_FROM

Tetrahedral Polyhedral Unfolding

NarratologiumENACTS

Decentered Global Perspectives

Academic Provenance: Narukawa, H. (2009)