Hyperbolic Geometry in Art

September 23, 2025

Hyperbolic Manuscripts

Mathematical Archaeology: Revealing Infinite Geometries in Historical Art

A research collection by Pardesco

Understanding Hyperbolic Geometry in Art

Hyperbolic geometry—where the angles of a triangle sum to less than 180 degrees and parallel lines diverge—remained a mathematical abstraction until M.C. Escher encountered it in 1958. Upon seeing H.S.M. Coxeter's diagrams, Escher experienced what he called "quite a shock": the realization that infinity could be captured within a finite circle.

This collection explores a revolutionary premise: that artists throughout history have intuited hyperbolic structures centuries before the mathematics existed to describe them. Through computational transformation, we can now reveal these hidden infinities.

Featured Work: Seven Stars to Infinity

Seven Stars to Infinity

Source: Embroidery Pattern with Seven Six-pointed Stars • Albrecht Dürer, before 1521

Submitted to JMM 2026 Art Exhibition

Seven Stars to Infinity - Hyperbolic transformation of Dürer

"I was struck by how Dürer's 1521 embroidery pattern—seven stars of unbroken knots—already suggested mathematical infinity through continuous strands. His Renaissance knotwork, influenced by Islamic geometry, contained a hidden truth. My transformation places his pattern in Poincaré disk space where six tiles meet at each vertex, creating endless recursion toward the boundary.

What makes this unique is the temporal dialogue: projecting 16th-century craftsmanship through 19th-century non-Euclidean geometry using 21st-century tools. The monochrome treatment strips away decoration to reveal pure mathematical structure—Dürer's vision of infinity finally realized through hyperbolic space he couldn't have calculated."

Historical Significance

Dürer's fascination with knot patterns emerged from his study of Leonardo da Vinci's "Academy" designs—complex interlaces that da Vinci himself had derived from Islamic geometric traditions. These patterns represented a mathematical puzzle that Renaissance artists approached intuitively: how to create infinite complexity from finite rules.

In 1521, Dürer created six woodcuts of intricate knot designs, including this pattern of seven six-pointed stars. While he understood these as exercises in continuous line drawing, he unknowingly encoded principles that wouldn't be mathematically formalized until Bolyai and Lobachevsky's work on non-Euclidean geometry three centuries later.

Mathematical Transformation: The {6,4} hyperbolic tessellation means six squares meet at each vertex, impossible in Euclidean space but natural in hyperbolic geometry. This creates a regular tiling that extends infinitely toward the disk boundary, where Dürer's finite pattern becomes infinite recursion.

Available: Limited edition museum-quality prints (50 total) at pardesco.com

The Escher Revolution and Beyond

Escher's Circle Limit series (1958-1960) represented the first deliberate artistic exploration of hyperbolic space. Working with mathematician H.S.M. Coxeter, Escher hand-calculated the positions of figures that would tessellate infinitely as they approached the boundary of his circular canvases.

His Circle Limit III, featuring fish in a {8,3} tessellation, required months of meticulous calculation. Each fish had to be drawn smaller and more precisely positioned as it approached the edge, where infinite fish swim at the boundary of the visible.

Today, computational tools allow us to go beyond Escher's pioneering work. Rather than creating new hyperbolic patterns, we can discover them within existing artworks—revealing that artists have always intuited these mathematical structures, encoding infinities they couldn't formally express.

The Complete Collection

The Infinite Emanations

Source: Chakrasamvara with His Consort Vajravarahi • Tibet, late 14th century

The tantric deities Chakrasamvara and Vajravarahi exist in eternal union, representing the inseparability of wisdom and compassion. Through {4,7} hyperbolic transformation—where seven squares meet at each vertex—their sacred mandala reveals its true nature: infinite emanations spiraling endlessly outward.

This 14th-century Tibetan thangka already encoded non-Euclidean principles. Buddhist cosmology describes infinite buddha-fields, each containing infinite buddhas. The {4,7} tessellation makes this theological concept mathematically visible—each emanation births seven more, approaching but never reaching the boundary of enlightenment.

Mathematical Note: The {4,7} tessellation is particularly significant as it cannot exist in Euclidean space—it requires negative curvature. This suggests that Tibetan artists intuited hyperbolic geometry through meditation on infinity, centuries before Gauss and Bolyai formalized these concepts.

Infinite Vajra

Source: Vajrabhairava Mandala • China, ca. 1330-32

This Yuan Dynasty kesi silk tapestry originally honored Emperor Tugh Temür, depicting Vajrabhairava—the buffalo-headed conqueror of death. The imperial commission required the finest materials: gold thread and silk in a technique so complex it was called "carved silk."

The [4,6] transformation reveals how Chinese imperial artists understood recursive space. Each chamber of the mandala contains the whole mandala; each protector deity guards infinite gateways. The golden pathways that once symbolized the emperor's divine mandate now extend infinitely, suggesting that power itself is fractal—endlessly subdividing yet maintaining its essential structure.

Art Historical Context: This mandala represents a unique Sino-Tibetan synthesis. Chinese aesthetic principles merge with Tibetan iconography, creating a hybrid form that already suggested mathematical transformation—two cultural geometries creating something beyond either tradition.

Surya's Infinite Chariot

Source: Mandala of the Sun God Surya by Kitaharasa • Nepal, 1379

One of only two dated 14th-century Nepalese paintings, this mandala by the artist Kitaharasa depicts Surya's eternal journey across the heavens. The original patron commissioned this work to expunge negative karma—seeking to break free from infinite cycles of cause and effect.

The [4,6] hyperbolic transformation reveals what the patron sought: escape from circular time into hyperbolic time, where parallel lines of karma diverge rather than return. Surya's seven horses, representing the days of the week, multiply infinitely—suggesting not repetition but endless variation, each day a new trajectory toward the unreachable boundary.

Named Artist Significance: Kitaharasa's signature makes this exceptionally rare. Most medieval Nepalese art remains anonymous, but here we can connect a specific consciousness to this mathematical intuition—one artist's vision of infinity preserved and now revealed through hyperbolic transformation.

"Each transformation is archaeological—not creating but discovering the hyperbolic geometries that artists encoded through spiritual and aesthetic intuition."

The Mathematical Process

Methodology: From Historical Artifact to Hyperbolic Space

Selection: Working exclusively with public domain masterworks that demonstrate geometric sophistication and cultural significance
Analysis: Identifying the underlying symmetry groups and applying appropriate hyperbolic functions ({4,6}, {4,7}, {6,4} tessellations)
Transformation: Using the Poincaré disk model to map infinite hyperbolic space into a finite circle—making the infinite visible and comprehensible
Preservation: Maintaining the essential character of the original while revealing its hidden mathematical nature
Documentation: Complete mathematical proofs and historical context accompany each transformation

Why This Matters: Reframing Art History Through Mathematics

This project suggests a radical reinterpretation of art history. If medieval and Renaissance artists intuited hyperbolic geometry centuries before its mathematical formalization, it implies that:

• Artistic intuition precedes mathematical formalization—artists discover through aesthetic exploration what mathematicians later prove
• Sacred geometry is inherently non-Euclidean—attempts to represent the infinite divine naturally lead to hyperbolic structures
• Cultural exchange encoded mathematical knowledge—Islamic geometry influenced European artists like Dürer, transmitting mathematical concepts through visual rather than symbolic language
• The computer becomes an archaeological tool—revealing layers of meaning invisible to previous generations