*Figure 1: Infinity Mirrored Room - Filled with the Brilliance of Life, 2011-17, Yayoi Kusama*

**Introduction**

Infinity is a widely-known yet enigmatic concept. Our first impression often refers to its mathematical definition–– an amount far beyond the scope of the humanly accountable or conceivable. Mathematicians have applied infinity as a fundamental idea in various contexts, from equating repeating decimals with fractions and calculating limits in calculus to projective geometry, set theory, fractals, and beyond. Despite the recent appearance of this idea, infinity has had a prominent presence in art and culture as early as philosophers from ancient Greece who contemplated about the endless. While mathematicians like Pythagoras observed that he could not measure lengths like the diagonal of a square (square root of 2) with a ruler with finite markings, theologians like Plotinus have challenged that the absolute being, whether God or simply an overarching mind encompassing all thought, is infinite rather than one. From here on, studies on infinity diverged into three branches: the mathematical, physical, and metaphysical. Artists have been a part of this ongoing conversation, exploring and experimenting with infinity in a broader sense: infinity as endless time or space, but also in emotion, expression, the cycle of life and death, and even in nothingness. By analyzing these fascinating works of art throughout history, many questions arise about the definition of infinity in art, the conceptual differences between a high degree of complexity and infinity, and whether the lack of anything is in itself a form of infinity. These diverse artistic interpretations, many alluding to scientific ideas and discoveries, speak volumes about how infinity fascinates interdisciplinary thinking and creativity.

**The Infinity Symbol**

The iconic "eight on its side" symbol, ∞, was invented in 1655 by the English mathematician John Wallis (who was the first to define infinity formally), who took inspiration from the roman numeral for 1000, ↀ. However, long before it was popularized as the universal infinity, the lemniscate shape appeared more than two millennia ago in ancient mythological art. Since the Egyptian Old Kingdom, the snake Uraeus or Ouraeus (figure 2) has often been depicted coiled in an endless ribbon-like pose. It symbolizes the protector serpent goddess Wadjet and has fittingly decorated every pharaoh's crown to exert eternal power.

*Figure 2: Various reliefs depicting Uraeus from ancient Egyptian mythology, Various dates, Unknown artists*

When the story of Uraeus eventually spread to the Greeks a few thousand years later, it was rewritten into the myth of Ouroboros (Greek for "tail-eating"). This snake (figure 3), coiled in the same manner but now eating its own tail, represents the cyclicality of life and death and the delicate balance between power and self-destruction. An enduring symbol in Western iconography, the Ouroboros establishes a visual association between the lemniscate and the forever-shifting balance of the universe.

*Figure 3: The Ouroboros, Unknown date, Unknown artist*

The ancient Romans built upon the myths of their predecessors by designing further complexity to the lemniscate form. One particular rendition added three-dimensionality to create the illusion of a twisted circular band. Similar shapes found on ancient mosaics established the evidence for one of the modern interpretations of the infinity symbol: a two-dimensional Möbius strip. The Möbius strip (discovered in 1858 almost simultaneously by Johann Benedict Listing and August Ferdinand Möbius) is a unique surface consisting of precisely one face and one edge (figure 6) and the fundamental component of all non-oriental (one face) surfaces. During the twentieth century, when there was a tendency for modern artists to respond to new scientific discoveries of their time, many sculptors created works reflecting their own vision of the Möbius strip. Robert R. Wilson, physicist and the first director of the Fermi National Accelerator Laboratory, created an identifiable yet novel, glittering sculpture (figure 4) by merging two Möbius bands at the edge.

*Figure 4: Möbius Band, 1974, Robert R. Wilson; Photo credits to Steve Krave (left) and Ken Wickham (right)*

American artist Charles O. Perry's *Continuum* further evolves the Möbius curve into a swirling bronze ribbon which he described as the "flow of matter through the center from positive to negative space and back again in a continuum." More interestingly, this work is not only symmetrical from its central axis but also from its northwest and southwest sides (the latter of which is shown in figure 5).

*Figure 5: Continuum, 1976, Charles O. Perry*

*Figure 6: Computerized models of a Möbius strip (left) and a Klein bottle (right)*

A few decades later, Carlo Séquin, a mathematical artist and a computer science professor at the University of California, Berkeley, took the Möbius loop to the next level (figure 7). He used Klein bottles (figure 7), another non-oriental surface with no edges from combining two Möbius strips symmetrically. Séquin used 3D printing to realize interweaving, tubular structures emphasizing form and line into a distinctive piece. One of the unique qualities of the Möbius strip is that one can run one continuous and infinite path along either its face or edge. Its petal-like twists, rhythmic turns, and cyclical completeness create a perfect, elegant, and symmetrical structure that naturally draws viewers into following its looped path and appeals to artists' desires to represent infinity.

*Figure 7: Dodecahedral Cluster of 25 Klein-Bottles, 2018, Carlo Séquin*

**Geometric Designs from Nature**

"You have endless subjects in nature," said David Hockney, and indeed, endlessness itself is a monumental subject of nature. There are many things, from the labyrinth-like pattern on the surface of coral reefs to the compact spirals of how sunflower seeds are composed together to the aerial view of fractal patterns of mountain ranges pushing against each other due to tectonic plates movement (figure 8). These captivating patterns found in nature have inspired both mathematicians and artists.

*Figure 8: Pattern found on coral reef is similar to space-filling curves in math (left); sunflower seeds grows in a spiral pattern which inspired numerous artists (middle), mountain ranges (Himalayan range) exhibits fractal patterns (right)*

*Figure 9: Hilbert's space-filling curves increasing in density*

In mathematical analysis, space-filling curves (figure 9) are curves whose range contains an entire two-dimensional lattice, most commonly a grid of squares or equilateral triangles. The ancient Greeks designed a pattern called the meander (figure 10) that decorated pottery, mosaics, and architectural friezes. This pattern, rolling like consecutive ocean waves, symbolizes the eternal and uninterrupted flow of human life. It is also a space-filling curve highly similar to the modern Hilbert's curves.

*Figure 10: Illustration of the Meander or Greek key, 1997, Lara Nagy*

Thousands of years later, software artist Don Relyea uses Hilbert's curve as the foundation for a series of abstract geometry, computer-generated artworks (figure 11). His algorithm recursively fills the canvas space with rectangles gradually decreasing in size, with the smaller rectangles centered on exact points on Hilbert's curve. Relyea's works not only embody some of the most intriguing results of program art but also the beauty of selection from a set of almost infinite possibilities.

*Figure 11: Cityscapes with Ladders and Helipads, 2004, Don Relyea*

Spirals are also of great interest to early thinkers, mainly because of their recurrent presence in the natural world, found in vastly different lifeforms, from seashells to succulents. The spiral is an infinitely extendable shape, and following it from its center outward in its pseudo-circular path holds a meditative quality. In the *Face of Christ on St. Veronica's Veil* (figure 12), the artist Dudesert uses a distinctive technique of unfolding the entire engraving with a single continuous, spiraling line. This alludes to both the threaded texture of St. Veronica's veil and the connection between the naturally powerful spiral and God's infinite love for humanity.

*Figure 12: Face of Christ on St. Veronica's Veil, 1700s, Dudesert*

In recent times, Stanford lecturer John Edmark is a sculptor and re-inventor of captivating spiral designs. His most well-known body of works consists of dome-like pieces of plastic that animate "never-ending blooms" (figure 13). Edmark sets his camera's shutter rate synchronized every time the sculpture makes a golden angle rotation of 137.5˚, derived from the golden ratio, a number closely related to producing natural spirals. His animations breathe life into still sculptures, showing spiral-based organisms gently swaying their soft tentacles in a back-and-forth motion.

*Figure 13: Bloom: Strobe-Animated Sculptures, 2014, John Edmark*

*Figure 14: The first six iterations of the Koch snowflake, one of the most famous fractals*

Rewind a few decades to the creative period of one of the most celebrated figures in mathematical art, M.C. Escher created tessellation patterns (figure 15) surprisingly accurate to fractals (figure 14) before the idea was rigorously defined in math. Fractals are geometric shapes that contain indefinitely detailed repetitions of their structure at arbitrary scales. The goldfishes gradually distort and shrink in size as it approaches the edge of the circle, with their spines aligned along the geodesics of the hyperbolic plane they occupy. With substantial foundations in mathematics, Escher creates a vision of a spherical dimension with infinite points and countless modes of rotational and reflective symmetry and one of the pioneering, enduring works of infinity.

*Figure 15: Circle Limit I, 1958, M.C. Escher*

**Infinite, Unaccountable Space-time**

There are three ways in which our world can appear unbounded or infinite. Time is forever. Space is limitlessly expandable. Any segment of space or time can be endlessly divided and subdivided. One of the first inspirations for the idea of an infinite universe is outer space. Since thousands of years ago, humans have been fascinated by the stars, which have formed our earliest calendars, instructed us in agriculture, and guided us to sail the seas. There is something transcending and magical about the dizzying, uncountable specks of light that have sparked our imagination about the infinite world beyond earth. Yayoi Kusama, one of the most influential contemporary artists today, has explored this idea in its rawest form through a series of infinity rooms (figure 16, 17). With *The Souls of Millions of Light Years*, Kusama creates the illusion of endless space consistent with our perception. Standing on a lone platform surrounded by warm and cool lights, the viewer feels as if floating on a plank on the dark waters of Styx towards the underworld.

*Figure 16: The Souls of Millions of Light Years, 2014, Yayoi Kusama*

Kusama transforms this design into another infinity room, Love Forever, with a completely different color concept and viewer experience. Polychromatic bulbs that line the ceiling are reflected across diagonally angled walls, along with a single opening in the wall for one to peer into the room. The yellowish-green atmosphere and the diamond lattice formed by the reflected space remind us of the view of an old amusement park at night as we stare into our own reflections in this hallucinogenic world. Whether it is being allowed to reside in the room itself or to look at it from a transparent window, Kusama forms an inseparable connection between the viewer and an "eternal unlimited universe" with "love for humanity, and longing for peace in the world."

*Figure 17: Love Forever, 1966-94, Yayoi Kusama*

A different interpretation of infinite space is one that can indefinitely expand. Artist Nicolas Baumgarten ingeniously uses digital tools to create *Zoomquilt*. Animated, this work of art slowly zooms into a detail in the picture, only to reveal an entirely distinct world, from the bridge to a lone tower to a dark forest of lizard-eating trees to an industrial complex to inside the mouth of an ogre to a floating city and back. The infinite cycle of scenes embedded within each other, strung together by a red ribbon, creates a surreal reality of falling into infinity. *Zoomquilt* reminds me of the film Inception, being a dream within a dream within a dream.

*Figure 18: Zoomquilt, 2004, Project created by Nicolas Baumgarten and illustrated by various artists*

With the latest developments in art, artificial intelligence has revolutionized how art is created. The assimilation of the multiverse theory into pop culture has ignited many's interests in the eternal existence of universes. In response, they have created art about their imaginations with A.I. tools (figure 19). Some incorporate vein-like energy currents (see bottom left image of figure 19) while others collage molten metal textures with planets (see bottom right image of figure 19), these unconventional works of art are the frontier to art on infinity today.

*Figure 19: Multiverse, 2021-22, Created by various users (@AstralIndica, @Ech0, @DDog345) using NightCafe Creator *

**Further Questions**

In this section, I want to present two questions that resulted from this research project. I will share some of my own thoughts, but the intention is that they are also left as food for thought for you, the reader.

**I. Can nothingness be a form of infinity?**

All the works we have examined have a degree of completeness or density of images. Installation artist Doug Wheeler, however, challenges this idea with Infinite Environment (figure 20), another infinity room in direct contrast with Kusama's creations. Is nothingness, in this case, a semi-circularly, entirely colorless space, a form of infinity? Does it evoke a sense of infinite space? Or does it allow us to access our infinite subconscious through meditation?

*Figure 20: Infinite Environment, 2012, Doug Wheeler*

**II. What is the conceptual difference between high complexity and infinity?**

Highly detailed works may all seem to convey the limitless, but some works make the conceptual distinction between the finite and the infinite. Artists who arduously decorated the Sheikh Lotfallah Mosque with arabesques (figure 21) at times included imperfections in their pattern purpose to convey the limitations and flaws of human imitation of God, who is the only truly infinite being. In what other instances or contexts do artists make these ideological choices?

*Figure 21: Details from the Sheikh Lotfallah Mosque, 1619, Baha' al-din al-'Amili, Ustad Mohammad Reza Isfahani (Architects)*

**Conclusion**

Since classical times, infinity has been a subject that has intrigued the brightest minds, from philosophers to mathematicians to physicists to artists. Ancient art about infinity is mostly symbolic or mythological, with inspirations rooted in nature and functions embedded in religion. Modern art about infinity, on the other hand, is an individualistic process that is far more conscious about the scientific backgrounds and technological art tools available. Today, infinite used in the context of art can mean all of these things, from the unaccountable to a looped time and cycles to profound spiritual or emotional realizations. It is a daunting concept. Infinity ultimately embodies the unknown in its purest form, a black hole pulling our interests and attention into something that we may never know fully about or may not exist at all. Nevertheless, human curiosity and creativity toward the idea of infinity, whether through a scientific or artistic lens, have blossomed into spectacular and diverse creations. Infinity is also an invaluable bridge between artists and scientists, two not -so-different sides of the same ever-thinking mind.

**Note from Author**

As a high school senior applying to college, this article is inspired by one of the University of Chicago's creative essay prompts, what represents the most worthwhile quality of humanity that you would share with a group of Martian explorers (paraphrased)? My response, built around Yayoi Kusama's *The Souls of Millions of Light Years Away*, allowed me to think about the relationship between artists and infinity for the first time, accumulating in this article. Finally, I thank my friend Minh for showing me an infinite painting, which further motivated me to write about this subject.

**Works Cited**

**General**

"Absolute (philosophy)." *Info:Main Page - New World Encyclopedia*, www.newworldencyclopedia.org/entry/absolute_(philosophy). Accessed 28 Nov. 2022.

Brecher, Kenneth. *The Bridges Archive*, archive.bridgesmathart.org/2017/bridges2017-153.pdf. Accessed 28 Nov. 2022.

"Infinity Symbol." *Wikipedia, the Free Encyclopedia*, Wikimedia Foundation, Inc, 15 Nov. 2022, en.wikipedia.org/wiki/Infinity_symbol. Accessed 28 Nov. 2022.

"Infinity | Definition, Symbol, & Facts." *Encyclopedia Britannica*, 10 Nov. 2022, www.britannica.com/science/infinity-mathematics#ref252427. Accessed 28 Nov. 2022.

"Infinity." *Wikipedia, the Free Encyclopedia*, Wikimedia Foundation, Inc, 15 Nov. 2022, en.wikipedia.org/wiki/Infinity. Accessed 28 Nov. 2022.

"Lemniscate." *Wikipedia, the Free Encyclopedia*, Wikimedia Foundation, Inc, 11 Jan. 2022, en.wikipedia.org/wiki/Lemniscate. Accessed 28 Nov. 2022.

Rucker, Rudy. "Infinity." *Mathematics at Dartmouth*, math.dartmouth.edu/~matc/Readers/HowManyAngels/InfinityMind/IM.html. Accessed 28 Nov. 2022.

**The Infinity Symbol**

**Geometric Designs from Nature**

**Infinite, Unaccountable Space-time**

**Further Exploration**

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