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Clifford A. Pickover
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Clifford Alan Pickover (born August 15, 1957) is an American author, editor, and columnist in the fields of science, mathematics, science fiction, innovation, and creativity. For many years, he was employed at the IBM Thomas J. Watson Research Center in Yorktown, New York, where he was editor-in-chief of the IBM Journal of Research and Development. He has been granted more than 700 U.S. patents, is an elected Fellow for the Committee for Skeptical Inquiry, and is author of more than 50 books, translated into more than a dozen languages.[1]

Key Information

Life, education and career

[edit]
Pickover's works have often dealt with higher dimensions, computer art, and visualization.

Pickover graduated first in his class from Franklin and Marshall College, after completing the four-year undergraduate program in three years. He received his PhD in 1982 from Yale University's Department of Molecular Biophysics and Biochemistry, where he conducted research on X-ray scattering and protein structure.[2]

Pickover was elected as a Fellow for the Committee for Skeptical Inquiry for his "significant contributions to the general public's understanding of science, reason, and critical inquiry through their scholarship, writing, and work in the media."[3] Other Fellows have included Carl Sagan and Isaac Asimov. He has been awarded almost 700 United States patents,[1] and his The Math Book was winner of the 2011 Neumann Prize.[4]

He joined IBM at the Thomas J. Watson Research Center in 1982, as a member of the speech synthesis group and later worked on the design-automation workstations.[5] For much of his career, Pickover has published technical articles in the areas of scientific visualization, computer art, and recreational mathematics.[2]

He is currently an associate editor for the scientific journal Computers and Graphics and is an editorial board member for Odyssey and Leonardo. He is also the Brain-Strain columnist for Odyssey magazine, and, for many years, he was the Brain-Boggler columnist for Discover magazine.

Pickover has received more than 100 IBM invention achievement awards, three research division awards, and four external honor awards.[2]

Work

[edit]
In the 1990s, Pickover created virtual caverns from extremely simple numerical simulations that reminded him of the Lechuguilla Cave, pictured here.[6][7]
Example of Pickover stalks in a detail of the Mandelbrot set

Pickover's primary interest is in finding new ways to expand creativity by melding art, science, mathematics, and other seemingly disparate areas of human endeavor.[8] In The Math Book and his companion book The Physics Book, Pickover explains that both mathematics and physics "cultivate a perpetual state of wonder about the limits of thoughts, the workings of the universe, and our place in the vast space-time landscape that we call home."[9] Pickover is an inventor with over 700 patents, the author of puzzle calendars, and puzzle contributor to magazines geared to children and adults. His Neoreality and Heaven Virus science-fiction series explores the fabric of reality and religion.[2]

Pickover is author of hundreds of technical papers in diverse fields, ranging from the creative visualizations of fossil seashells,[10] genetic sequences,[11][12] cardiac[13] and speech sounds, and virtual caverns[14] and lava lamps,[15] to fractal and mathematically based studies.[16][17][18][19] He also has published articles in the areas of skepticism (e.g. ESP and Nostradamus), psychology (e.g. temporal lobe epilepsy and genius), and technical speculation (e.g. "What if scientists had found a computer in 1900?" and "An informal survey on the scientific and social impact of a soda can-sized super-super computer").[20] Additional visualization work includes topics that involve breathing motions of proteins,[21] snow-flake like patterns for speech sounds,[22] cartoon-face representations of data,[23] and biomorphs.[24]

Pickover has also written extensively on the reported experiences of people on the psychotropic compound DMT.[25][26] Such apparent entities as Machine Elves are described as well as "Insects From A Parallel Universe".[26]

On November 4, 2006, he began Wikidumper.org, a popular blog featuring articles being considered for deletion by English Wikipedia.

Pickover stalks

[edit]

Pickover stalks are certain kinds of details that are empirically found in the Mandelbrot set in the study of fractal geometry. In the 1980s, Pickover proposed that experimental mathematicians and computer artists examine the behavior of orbit trajectories for the Mandelbrot set in order to study how closely the orbits of interior points come to the x and y axes in the complex plane. In some renditions of this behavior, the closer that the point approaches, the higher up the color scale, with red denoting the closest approach. The logarithm of the distance is taken to accentuate the details. This work grew from his earlier work with Julia sets and "Pickover biomorphs," the latter of which often resembled microbes.[27][28]

Frontiers of Scientific Visualization

[edit]

In "Frontiers of Scientific Visualization" (1994) Pickover explored "the art and science of making the unseen workings of nature visible". The books contains contributions on "Fluid flow, fractals, plant growth, genetic sequencing, the configuration of distant galaxies, virtual reality to artistic inspiration", and focuses on use of computers as tools for simulation, art and discovery.[29]

Visualizing Biological Information

[edit]

In "Visualizing Biological Information" (1995) Pickover considered "biological data of all kinds, which is proliferating at an incredible rate". According to Pickover, "if humans attempt to read such data in the form of numbers and letters, they will take in the information at a snail's pace. If the information is rendered graphically, however, human analysts can assimilate it and gain insight much faster. The emphasis of this work is on the novel graphical and musical representation of information containing sequences, such as DNA and amino acid sequences, to help us find hidden pattern and meaning".[30]

Vampire numbers and other mathematical highlights

[edit]

In mathematics, a vampire number or true vampire number is a composite natural number v, with an even number of digits n, that can be factored into two integers x and y each with n/2 digits and not both with trailing zeroes, where v contains all the digits from x and from y, in any order. x and y are called the fangs. As an example, 1260 is a vampire number because it can be expressed as 21 × 60 = 1260. Note that the digits of the factors 21 and 60 can be found, in some scrambled order, in 1260. Similarly, 136,948 is a vampire because 136,948 = 146 × 938.

Vampire numbers first appeared in a 1994 post by Clifford A. Pickover to the Usenet group sci.math, and the article he later wrote was published in chapter 30 of his book Keys to Infinity.[31]

In addition to "Vampire numbers",[32] a term Pickover actually coined, he has coined the following terms in the area of mathematics: Leviathan number,[33] factorion,[34] Carotid–Kundalini function and fractal,[35] batrachion,[36] Juggler sequence,[37] and Legion's number,[38] among others. For characterizing noisy data, he has used Truchet tiles and Noise spheres,[39] the later of which is a term he coined for a particular mapping, and visualization, of noisy data to spherical coordinates.

In 1990, he asked "Is There a Double Smoothly Undulating Integer?",[40] and he computed "All Known Replicating Fibonacci Digits Less than One Billion".[41] With his colleague John R. Hendricks, he was the first to compute the smallest perfect (nasik) magic tesseract.[42] The "Pickover sequence" dealing with e and pi was named after him,[43] as was the "Cliff random number generator"[44] and the Pickover attractor, sometimes also referred to as the Clifford Attractor.[45][46]

Culture, religion, belief

[edit]

Starting in about 2001, Pickover's books sometimes began to include topics beyond his traditional focus on science and mathematics. For example, Dreaming the Future discusses various methods of divination that humans have used since stone-age times. The Paradox of God deals with topics in religion. Perhaps the most obvious departure from his earlier works includes Sex, Drugs, Einstein, and Elves: Sushi, Psychedelics, Parallel Universes, and the Quest for Transcendence, which explores the "borderlands of science" and is "part memoir and part surrealist perspective on culture.".[47] Pickover follows-up his "quest for transcendence" and examination of popular culture with A Beginner's Guide to Immortality: Extraordinary People, Alien Brains, and Quantum Resurrection.

History of science and mathematics

[edit]

Starting in 2008, Pickover's books began to focus on the history of science and mathematics, with such titles as Archimedes to Hawking, as well as The Math Book, The Physics Book, and The Medical Book—a trilogy of more than 1,500 pages that presents various historical milestones, breakthroughs, and curiosities.

WikiDumper.org

[edit]

Wikidumper.org is a website created by Pickover that promises to permanently record a snapshot of the "best of the English Wikipedia rejects", articles that are slated for deletion at the English Wikipedia. WikiDumper was launched on November 4, 2006, and accepts user submissions. Although the site doesn't specify its criteria for inclusion, many of its articles don't cite their sources. The site has been criticized as likely to be less accurate than English Wikipedia.[48]

Publications

[edit]
Visualization of chaotic attractor. Pickover's earliest books often focused on patterns that characterize mathematics such as fractals, chaos, and number theory. Computer graphics, reminiscent of this chaotic attractor, were common in his early works.
Forest troll. (Theodor Kittelsen, 1906). Some of Pickover's later books often discussed "science at the edges," including such topics as parallel universes, quantum immortality, alien life, and elf-like beings seen by some people who use dimethyltryptamine.

Pickover is author of over forty books on such topics as computers and creativity, art, mathematics, black holes, human behavior and intelligence, time travel, alien life, Albert Einstein, religion, dimethyltryptamine elves, parallel universes, the nature of genius, and science fiction.[49][50]

Books

[edit]
  • 1990. Computers, Pattern, Chaos, and Beauty. St. Martin's Press. ISBN 0-486-41709-3
  • 1991. Computers and the Imagination. St. Martin's Press.
  • 1992. Mazes for the Mind. St. Martin's Press.
  • 1994. Chaos in Wonderland. St. Martin's Press.
  • 1995. Keys to Infinity. Wiley.
  • 1996. Black Holes: A Traveler's Guide. Wiley.
  • 1997. The Alien IQ Test. Basic Books.
  • 1997. The Loom of God. Plenum.
  • 1998. Spider Legs. With Piers Anthony TOR.
  • 1998. The Science of Aliens. Basic Books.
  • 1998. Time: A Traveler's Guide. Oxford University Press.
  • 1999. Strange Brains and Genius: The Secret Lives of Eccentric Scientists and Madmen, Harper Perennial/Quill, ISBN 0-688-16894-9
  • 1999. Surfing Through Hyperspace. Oxford University Press.
  • 2000. Cryptorunes: Codes and Secret Writing. Pomegranate.
  • 2000. The Girl Who Gave Birth to Rabbits. Prometheus.
  • 2000. Wonders of Numbers. Oxford University Press.
  • 2001. Dreaming the Future. Prometheus.
  • 2001. The Stars of Heaven. Oxford University Press.
  • 2002. The Zen of Magic Squares, Circles, and Stars. Princeton University Press. ISBN 0-691-11597-4
  • 2002. The Mathematics of Oz. Cambridge University Press. ISBN 0-521-01678-9
  • 2002. The Paradox of God and the Science of Omniscience. St. Martin's Press. ISBN 1-4039-6457-2
  • 2003. Calculus and Pizza. John Wiley & Sons. ISBN 0-471-26987-5
  • 2005. Sex, Drugs, Einstein, and Elves. Smart Publications. ISBN 1-890572-17-9
  • 2005. A Passion for Mathematics, John Wiley & Sons. ISBN 0-471-69098-8
  • 2006. The Mobius Strip, Thunder's Mouth Press. ISBN 1-56025-826-8
  • 2007. A Beginner's Guide to Immortality. Thunder's Mouth Press. ISBN 978-1-56025-984-8
  • 2007. The Heaven Virus. Lulu. ISBN 978-1-4303-2969-5
  • 2008. Archimedes to Hawking: Laws of Science and the Great Minds Behind Them. Oxford University Press. ISBN 978-0-19-533611-5
  • 2009. Jews in Hyperspace. Kindle Edition.
  • 2009. The Loom of God. Sterling Publishing. ISBN 978-1-4027-6400-4
  • 2009. The Math Book: From Pythagoras to the 57th Dimension, 250 Milestones in the History of Mathematics. Sterling Publishing. ISBN 978-1-4027-5796-9
  • 2011. The Physics Book: From the Big Bang to Quantum Resurrection. Sterling Publishing. ISBN 978-1-4027-7861-2
  • 2012. The Medical Book: From Witch Doctors to Robot Surgeons. Sterling Publishing. ISBN 978-1-4027-8585-6
  • 2012. Brain Strain: A Mental Muscle Workout That's Fun!. Cricket Media.
  • 2013. The Book of Black: Black Holes, Black Death, Black Forest Cake and Other Dark Sides of Life, Calla Editions. ISBN 978-1606600498
  • 2014. The Mathematics Devotional: Celebrating the Wisdom and Beauty of Mathematics. Sterling Publishing. ISBN 978-1454913221.
  • 2015. The Physics Devotional: Celebrating the Wisdom and Beauty of Physics. Sterling Publishing . ISBN 978-1454915546
  • 2015. Death and the Afterlife: A Chronological Journey, from Cremation to Quantum Resurrection. Sterling Publishing. ISBN 978-1454914341
  • 2018. The Science Book: From Darwin to Dark Energy. Sterling Publishing. ISBN 978-1454930068
  • 2019. Artificial Intelligence: An Illustrated History, From Medieval Robots to Neural Networks. Sterling Publishing. ISBN 978-1454933595
  • Mind-Bending Puzzles (calendars & cards), Pomegranate, each year

Neoreality science fiction series

[edit]
  • 2002. Liquid Earth. The Lighthouse Press, Inc.
  • 2002. The Lobotomy Club. The Lighthouse Press, Inc.
  • 2002. Sushi Never Sleeps. The Lighthouse Press, Inc.
  • 2002. Egg Drop Soup. The Lighthouse Press, Inc. ISBN 0-9714827-9-9

Edited collections

[edit]
  • 1992. Spiral Symmetry, World Scientific. ISBN 981-02-0615-1
  • 1993. Visions of the Future: St. Martin's Press.
  • 1994. Frontiers of Scientific Visualization. Wiley.
  • 1995. Future Health: Computers & Medicine in the 21st Century. St. Martin's Press.
  • 1995. The Pattern Book: Fractals, Art, and Nature. World Scientific.
  • 1995. Visualizing Biological Information. World Scientific.
  • 1996. Fractal Horizons. St. Martin's Press,
  • 1998. Chaos and Fractals. Elsevier.

See also

[edit]

References

[edit]
[edit]
Revisions and contributorsEdit on WikipediaRead on Wikipedia

Clifford A. Pickover

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Clifford A. Pickover (born 1957) is an American author, mathematician, and inventor renowned for his interdisciplinary explorations at the intersection of science, mathematics, art, and creativity. With a Ph.D. in molecular biophysics and biochemistry from Yale University and a first-in-class graduation in three years from Franklin and Marshall College, Pickover has built a distinguished career as a research staff member at IBM's . There, he has been granted over 830 U.S. patents—many owned by IBM and other companies like Google—and received more than 100 invention achievement awards, three research division awards, and four external honors. He also served for many years as Editor-in-Chief of the and has held editorial roles, including associate editor for Computers and Graphics for about 20 years and board member for journals like Leonardo and Odyssey. Pickover's prolific output includes over 50 books, translated into dozens of languages, covering diverse topics from mathematical milestones and physics wonders to time travel, religion, and science fiction. Notable works include , 250 Milestones in the History of Mathematics (2009), which won the 2011 Neumann Prize from the British Society for the History of Mathematics for its contributions to the history of mathematics; (2011); and Time: A Traveler's Guide (1998). He has co-authored books such as Spider Legs (1998) with Piers Anthony and contributed over 200 articles to publications like Scientific American, Harper's Magazine, and Wired. His innovations include the 3-D computer mouse (U.S. Patent 5,095,302) and fractal visualizations inspired by the Mandelbrot set. Among his recognitions, Pickover is an elected Fellow of the and won first prize in the Institute of Physics' "Beauty of Physics Photographic Competition." His work has garnered praise in outlets like the Los Angeles Times, which called him "a perpetual idea machine," and The New York Times, highlighting his ability to make complex ideas accessible. Pickover's contributions continue to inspire wonder by melding disparate fields, as evidenced by his ongoing research and writing on creativity and innovation.

Biography

Early Life and Education

Clifford A. Pickover was born on August 15, 1957, in Ocean, New Jersey, to Merwin B. Pickover, an electrical engineer, and Regina Taubman, a teacher. From an early age, Pickover exhibited a fascination with mathematics, science, and puzzles, influenced by his family's environment that provided exposure to computers through his father's engineering work and to creative pursuits via his mother's teaching career. Pickover graduated first in his class from Franklin and Marshall College in 1978, earning a B.A. summa cum laude in physics and mathematics after completing the four-year program in just three years. He went on to pursue graduate studies at Yale University, where he received an M.Phil. in 1980 and a Ph.D. in molecular biophysics and biochemistry in 1982, with his dissertation focusing on computational modeling of protein structures via X-ray scattering methods. Pickover's initial research interests centered on biophysics, resulting in early publications such as his 1979 co-authored paper in the Journal of Biological Chemistry examining substrate binding and domain closure in yeast phosphoglycerate kinase, contributing to understanding protein conformational changes.

Personal Influences and Awards

Clifford A. Pickover's intellectual pursuits have been profoundly shaped by literary figures who masterfully intertwined mathematics, art, and speculative fiction, including Martin Gardner, Lewis Carroll, and Jorge Luis Borges. Gardner, renowned for his recreational mathematics columns in Scientific American, influenced Pickover's approach to making complex ideas accessible and entertaining. Carroll's whimsical explorations of logic and absurdity in Alice's Adventures in Wonderland resonated with Pickover's interest in paradoxical structures, which he references in his writings on mathematical curiosities. Borges's labyrinthine tales, like those in Ficciones, inspired Pickover's blending of infinite geometries with philosophical mysticism, as seen in his discussions of hyperspace and infinity. Pickover's fascination with creativity, innovation, and the convergence of science and mysticism stems from his personal explorations into altered states of consciousness, including psychedelics and dreams. These experiences informed his worldview, emphasizing the boundaries between rational inquiry and transcendent perception. His 2005 book Sex, Drugs, Einstein, and Elves: Sushi, Psychedelics, Parallel Universes, and the Quest for Transcendence serves as an autobiographical lens into these themes, presenting a collection of personal essays that reflect on Mediterranean journeys and encounters with substances like , framing them as catalysts for innovative thinking at the intersection of physics, biology, and the esoteric. Pickover is married to Elahe Khorasani, a computer specialist. Among Pickover's notable accolades is the 2011 Neumann Prize, awarded by the British Society for the History of Mathematics for The Math Book, recognizing its exemplary contribution to popularizing mathematical history for a broad audience. In 2005, he was elected a Fellow of the Committee for Skeptical Inquiry (CSI) for his significant efforts in promoting scientific understanding and critical thinking through authorship and editorial work. At IBM, where he holds Master Inventor status—the company's highest recognition for innovation—Pickover has amassed over 830 U.S. patents as of 2025, underscoring his impact on computational visualization and user interfaces. Pickover's contributions to scientific visualization and public education have earned further acclaim, particularly through his long-running "Brain-Boggler" column in Discover magazine, which popularized puzzles and concepts in mathematics and science for general readers from the 1990s onward. This work, alongside his editorial roles, highlights his role in bridging esoteric knowledge with everyday curiosity, fostering innovation across disciplines.

Professional Career

IBM Research and Inventions

Clifford A. Pickover joined IBM's Thomas J. Watson Research Center in 1982 as a research staff member in the speech synthesis group, where he initially focused on advancing audio processing technologies. Over the subsequent decades, he transitioned to contributions in design-automation workstations and contributed to the promotion of IBM's IntelliStation, a high-end personal computer line that integrated advanced computational capabilities for scientific and engineering applications, by creating demonstrations and applications. Pickover advanced to prominent roles within IBM, earning recognition as a Master Inventor and achieving the highest invention plateau at the Watson Research Center, reflecting his sustained impact on technological innovation. He has received more than 100 IBM Invention Achievement Awards, along with three Research Division Awards, underscoring his leadership in interdisciplinary teams that applied mathematical principles to engineering challenges. As of 2025, Pickover holds over 830 U.S. patents, the majority assigned to IBM, with a focus on user interfaces, data visualization, computer graphics, and AI-driven applications. His inventive work extended to computational biology and pattern recognition, including the development of software for molecular modeling and structural characterization of biomolecules, which facilitated advanced simulations in drug design and protein analysis. For example, in a 1987 IBM Technical Disclosure, Pickover detailed a software package enabling interactive visualization and manipulation of molecular structures to support research in biophysics. In pattern recognition, his contributions included algorithms for analyzing chaotic networks, enhancing IBM's capabilities in signal processing and predictive modeling. Additionally, patents such as U.S. Patent 9,177,257 for real-time cognitive assistance systems demonstrate his innovations in AI interfaces that provide proactive user support. Pickover's patents on interactive educational tools, like U.S. Patent 6,505,208 for monitoring and evaluating learning methods via computing devices, have influenced adaptive training technologies. These inventions emphasize interdisciplinary fusion, such as fractal-based techniques for image processing in graphics and visualization tools for biological data.

Editorial Roles and Collaborations

Clifford A. Pickover served as editor-in-chief of the IBM Journal of Research and Development for many years, during which he oversaw the publication of articles on advancements in computing, innovation, and related scientific fields. He has held the position of associate editor for the journal Computers & Graphics, contributing to its focus on computer-generated imagery, visualization techniques, and interdisciplinary graphics research. Pickover is also a member of the editorial board for Odyssey magazine, which explores science and mathematics for general audiences, and for Leonardo, a publication dedicated to the intersections of art, science, and technology. For many years, Pickover wrote the "Brain Boggler" column for Discover magazine, where he created puzzles that integrated scientific concepts with creative problem-solving to engage readers in mathematics and physics.

Mathematical Contributions

Fractal Geometry: Pickover Stalks

In the 1980s, Clifford A. Pickover introduced as intricate visual details within the Mandelbrot set, resembling elongated stalks or antennae that highlight the set's boundary complexities in fractal geometry. These features arose from Pickover's innovative rendering techniques during his research at IBM's Thomas J. Watson Research Center, where he explored mathematical feedback loops to generate biomorph-like patterns from chaotic iterations. The core mathematical formulation of Pickover stalks builds on the Mandelbrot set's defining iteration zn+1=zn2+cz_{n+1} = z_n^2 + c, with z0=0z_0 = 0 and cc a complex parameter. Unlike standard escape-time rendering, Pickover stalks employ an orbit trap method using the real and imaginary axes as linear traps; for each cc, the orbit's points are iterated until divergence or a maximum step, and the pixel color is assigned based on the minimum distance of any orbit point to these axes, accentuating filamentary structures where orbits graze the traps. These stalks manifest prominently in detailed boundary regions of the Mandelbrot set, such as antennae-like appendages, revealing hidden dynamical behaviors through parametric tracing of orbit minima. Pickover's early visualizations of these stalks, computed on IBM hardware like the PS/2 workstation, advanced computer graphics by enabling high-resolution depictions of fractal intricacies and contributed to chaos theory by illustrating how sensitive initial conditions produce self-similar, biologically evocative forms. The technique's applications extended to modeling irregular patterns in natural phenomena, bridging abstract mathematics with visual aesthetics. The concept of Pickover stalks evolved significantly in Pickover's 1990 book Computers, Pattern, Chaos, and Beauty, where full-color plates and discussions emphasized their role in uncovering "unseen worlds" of complexity, influencing subsequent fractal art and scientific illustration.

Number Theory: Vampire Numbers

Clifford A. Pickover introduced the concept of in a post to the Usenet group sci.math on October 31, 1994. He coined the term to describe a playful curiosity in recreational mathematics, inspired by the eerie allure of vampires in literature. A vampire number is a composite natural number with an even number of digits, denoted as nn, that can be expressed as the product of two integers, each with n/2n/2 digits, such that the multiset of digits in the two factors forms a permutation of the digits in the original number, excluding any trailing zeros in the factors unless the original number ends in an even number of zeros. The two factors are termed "fangs," and neither can have leading zeros. Pickover detailed this definition in chapter 30 of his book Keys to Infinity, published in 1995, where he expanded on the idea with additional examples and properties. Trailing zeros are permitted in the fangs only if they appear in pairs (one in each factor) to match those in the vampire number, ensuring the digit permutation holds without artificial padding. Key properties include the requirement for an even digit count in the vampire number and the strict digit permutation condition, which distinguishes vampire numbers from mere factorizations. Variants exist, such as "fanged" vampire numbers, which admit multiple distinct fang pairs satisfying the conditions, and "unfanged" ones limited to a single pair. Pseudovampire numbers, also discussed by Pickover, relax the equal-digit-length rule for fangs but retain the digit permutation property. Representative examples of four-digit vampire numbers include 1260, which factors as 21×6021 \times 60 (digits 1,2,6,0 permute to match), and 1395 as 15×9315 \times 93 (digits 1,3,9,5). Larger examples, such as the eight-digit 10025010 (2001×50102001 \times 5010), illustrate the concept's scalability, though such numbers become rarer as digit length increases. To generate vampire numbers, one can iterate over candidate composites with even digit lengths, identify all factor pairs where each factor has exactly half the digits, and verify if the combined digits of the pair (ignoring leading zeros and handling trailing zeros appropriately) form a permutation of the candidate's digits. This algorithmic approach, suitable for computational enumeration, has been used to catalog sequences like the seven known four-digit vampires, the first seven terms of the Online Encyclopedia of Integer Sequences (OEIS A014575). Pickover's invention has popularized vampire numbers in mathematical puzzles and recreational contexts, inspiring challenges to find larger instances or variants. It has led to extensions like multimorphic numbers, which generalize fanged vampires to allow multiple fang pairs across varying digit configurations.

Other Innovations in Mathematics

Pickover contributed to the exploration of L-systems, formal grammar systems originally developed by Aristid Lindenmayer for modeling plant growth, by editing and promoting their application in fractal geometry for generating plant-like structures and algorithmic art. In his 1996 edited volume Fractal Horizons: The Future Use of Fractals, a dedicated chapter details L-systems as string-rewriting mechanisms to produce branching patterns resembling trees and foliage, emphasizing their role in simulating natural morphology through iterative rules such as axiom F and productions like F → FF+[+F-F-F]-[-F+F+F]. Pickover extended these concepts to three-dimensional rendering, integrating L-systems with computer graphics techniques to create volumetric fractal models of organic forms, as showcased in his 1991 book Fractal 3D Magic, where such methods visualize complex, self-similar structures in spatial environments. In Wonders of Numbers: Adventures in Mathematics, Mind, and Meaning (2001), Pickover advanced popular understanding of π approximations and infinite series through visual and computational explorations, highlighting feats like the 1998 calculation of the five-trillionth binary digit of π (a 0) using distributed computing across 25 machines, which underscores the transcendental number's infinite non-repeating nature and its ties to series expansions such as the Leibniz formula. He emphasized mnemonic devices for memorizing π digits, presenting them as creative tools to encode sequences like 3.14159 into memorable phrases or patterns, fostering intuitive grasp of infinite series convergence without deep algebraic derivation. In the 2020s, Pickover incorporated AI into mathematical inquiry, particularly in puzzle-solving and pattern recognition, as explored in his 2024 book Artificial Intelligence: An Illustrated History—From Medieval Robots to Neural Networks, where he connects historical AI concepts to modern applications in generating and solving math puzzles, such as neural networks approximating irrational numbers or optimizing series visualizations. This work ties AI tools to traditional mathematical recreation, enabling automated discovery of novel approximations and fractal extensions in three dimensions.

Visualization and Art

Scientific Visualization Techniques

Clifford A. Pickover advanced during his tenure as a research staff member at IBM's T. J. Watson Research Center, where he focused on rendering techniques to depict complex, abstract scientific data. In the edited volume Frontiers of Scientific Visualization (1994), co-edited with Stuart K. Tewksbury and published by Wiley, Pickover presented pioneering applications of ray-tracing and volume rendering for visualizing unseen phenomena such as fluid flows and chaotic systems. These methods enabled the creation of photorealistic images and volumetric displays from multidimensional datasets, bridging computational simulation with intuitive graphical representation to aid scientific discovery. A key aspect of Pickover's approach involved mapping higher-dimensional data onto 2D and 3D visuals to make abstract structures accessible. For instance, in his 1988 paper "Visualization of Quaternion Slices," published in Image and Vision Computing, he introduced techniques for slicing and projecting four-dimensional quaternion spaces into lower dimensions, using iterative projections and shading to reveal geometric properties otherwise hidden in higher realms. Similarly, he employed color mappings to encode dynamic behaviors in chaos attractors, assigning hues based on iteration counts or basin memberships to highlight trajectories and basins of attraction in nonlinear systems, as explored in Computers, Pattern, Chaos, and Beauty (1990). These color schemes, often generated via lookup tables, transformed iterative computations into vibrant, interpretable graphics that emphasized the aesthetic and analytical value of chaotic dynamics. Pickover's IBM collaborations in the 1980s and 1990s extended these techniques into virtual reality prototypes for scientific simulations, leveraging hardware like the IBM POWER Visualization System. In a 1994 paper in Computers & Graphics, he described parallel processing methods for generating aeolian fractals—simulations of wind-eroded landscapes—on this system, enabling real-time rendering of large-scale environmental models for geoscientific analysis. These VR efforts integrated stereoscopic displays and interactive navigation, allowing researchers to immerse in simulated physical processes such as turbulence, thereby enhancing hypothesis testing and data exploration. Such prototypes foreshadowed modern immersive tools, demonstrating Pickover's emphasis on interactivity in scientific insight. In 1986, Pickover developed the (SDP), a visualization technique that transforms one-dimensional time series data—such as speech waveforms or other sampled signals—into two-dimensional polar symmetric scatter plots. This method highlights hidden patterns and subtle acoustic or dynamic differences that are otherwise difficult to discern visually. SDP has generated substantial follow-up research, with over 100 scholarly citations and applications in domains including predictive maintenance, condition monitoring, acoustic classification, and fusion plasma studies. Its low computational cost and compatibility with modern deep-learning pipelines, such as using SDP images as inputs to convolutional neural networks, maintain its relevance, as evidenced by publications into the mid-2020s. As an example of applied methodology, Pickover's visualization techniques have been used to render structures like Pickover stalks, where 3D projections and color gradients illustrate bifurcations in parametric equations. Overall, these contributions prioritized conceptual clarity, using selective metrics like rendering speed on IBM systems to underscore practical impact without exhaustive benchmarks.

Biological Information Visualization

Clifford A. Pickover advanced the field of biological information visualization through innovative graphical methods for representing genomic data, particularly DNA sequences. In his 1987 paper, he introduced "DNA vectorgrams," a technique that maps nucleotide sequences as directed movements on a two-dimensional cellular lattice, transforming linear genetic information into visual patterns that highlight structural anomalies, such as those in cancer genes. This approach facilitates pattern recognition by converting abstract sequence data into interpretable geometric paths, aiding researchers in identifying repetitions or irregularities without exhaustive numerical analysis. Pickover's 1995 edited volume, Visualizing Biological Information, further developed these ideas by compiling methods for DNA sequence mapping, including fractal-based functions derived from nucleotide compositions to generate self-similar visual landscapes that reveal hidden periodicities and motifs. The book emphasizes graph-theoretic representations, such as lattice plots and spectral analyses, to compress and display large genomic datasets, enabling biologists to detect evolutionary relationships or functional elements more intuitively than through traditional sequence alignments. Complementary techniques in the volume address amino acid sequences, using graphical embeddings to visualize protein primary structures as navigable diagrams. Building on his PhD research in molecular biophysics and biochemistry at Yale University, completed in 1982 with a focus on X-ray scattering for protein structure determination, Pickover's later visualizations included recursive algorithms that produce tree-like and helical structures mimicking protein folding pathways, often enhanced with color-coding to denote conformational errors or energy states in molecular dynamics. These methods, exemplified in his 1989 paper on fractal simulations of biological shapes, prioritize conceptual clarity over computational intensity, allowing for rapid prototyping of biomolecular models.

Cultural and Philosophical Explorations

Religion, Belief, and Spirituality

Pickover's explorations into the intersections of science, mysticism, and human belief often draw from personal experiences, blending empirical inquiry with philosophical reflection. In his 2005 book Sex, Drugs, Einstein, and Elves: Sushi, Psychedelics, Parallel Universes, and the Quest for Transcendence, he delves into altered states of consciousness induced by psychedelics such as DMT, ayahuasca, and LSD, recounting encounters with entities like "machine-elves" and insectile beings during these experiences. Pickover connects these visions to dreams, lucid dreaming, and near-death experiences, positing them as potential portals to parallel realities and enhanced creativity, while incorporating cultural and scientific perspectives on transcendence. The work combines memoir with analysis, emphasizing how such states challenge conventional notions of reality and spirituality. Through a mathematical lens, Pickover examines sacred geometry and its role in religious traditions in The Zen of Magic Squares, Circles, and Stars: An Exhibition of Surprising Structures Across Dimensions (2002). He traces the historical use of , circles, and stars in ancient cultures, including China, Babylon, and Mesoamerica, where these patterns served mystical purposes such as warding off evil and invoking good fortune. The book highlights comparative aspects across belief systems, illustrating how geometric constructions embody spiritual symbolism and cosmic order, from talismans in Islamic and Hindu practices to protective amulets in Judeo-Christian contexts. Pickover's analysis underscores the harmony between mathematics and mysticism, presenting these structures as bridges between rational inquiry and transcendent wonder. In (2015), Pickover adopts a skeptical yet open-minded approach to inquiries about the afterlife, integrating scientific concepts like quantum immortality with philosophical and cultural narratives. He surveys historical beliefs—from Maya death gods and golems to séances and zombies—while exploring speculative ideas such as quantum resurrection, where consciousness might persist through multiverse branching. This work balances empirical caution with curiosity about quantum consciousness, questioning whether scientific frameworks can illuminate enduring human quests for meaning beyond death. In The Paradox of God and the Science of Omniscience (2001), Pickover explores logical paradoxes related to divine omniscience, employing puzzles, brainteasers, and vignettes that intersect theology, philosophy, science, and logic. The book examines the implications of an all-knowing God for concepts like free will and reality, drawing from biblical traditions to quantum physics and encouraging reflection on the limits of knowledge and belief.

History of Science and Mathematics

Clifford A. Pickover has contributed significantly to the popular understanding of the history of science and mathematics through illustrated chronological narratives that highlight key milestones, influential figures, and broader cultural impacts. In Archimedes to Hawking: Laws of Science and the Great Minds Behind Them (2008), he examines over 40 eponymous physical laws spanning from Archimedes' Principle in 250 B.C. to Stephen Hawking's contributions in the 1970s, providing biographical insights into the discoverers and the historical contexts that shaped their work. This book emphasizes the human elements behind scientific progress, including the diverse backgrounds of polymaths like Isaac Newton and Marie Curie, and how societal environments influenced breakthroughs in areas such as motion, electromagnetism, and cosmology. Pickover extends this approach to mathematics in The Math Book: From Pythagoras to the 57th Dimension, 250 Milestones in the History of Mathematics (2009, revised 2025), which chronicles 250 pivotal developments from ancient "ant odometers" around 150 million B.C. to modern concepts like the mathematical universe hypothesis in 2007, including advancements up to quantum computing in the revised edition. The work places particular emphasis on overlooked contributors, such as female mathematicians Hypatia, Maria Gaetana Agnesi, and Sofia Kovalevskaya, and integrates cultural contexts, notably the Islamic Golden Age with milestones like Muhammad ibn Musa al-Khwarizmi's foundational algebra in 830 and the Indian text Ganita Sara Samgraha in 850, which advanced arithmetic and geometry amid a flourishing of non-European scholarship. These entries, accompanied by full-color illustrations and explanations of formulas and applications, underscore how mathematical ideas evolved across civilizations, often bridging philosophy, art, and science. Similarly, in The Physics Book: From the Big Bang to Quantum Resurrection, 250 Milestones in the History of Physics (2011, revised 2025 edition), Pickover traces the discipline's evolution from the Big Bang approximately 13.7 billion years ago through ancient inventions like the sundial (3000 B.C.) and Baghdad batteries (250 B.C.) to speculative futures like quantum resurrection beyond 100 trillion years. The book highlights paradigm-shifting events, such as Newton's laws of motion and gravitation (1687), Einstein's general theory of relativity (1915), and the discovery of X-rays (1895), while contextualizing them within historical challenges faced by figures like James Clerk Maxwell and Erwin Schrödinger. Addressing contemporary gaps in historical coverage, Pickover's Artificial Intelligence: An Illustrated History: From Medieval Robots to Neural Networks (2024) fills voids in mainstream accounts by detailing AI's trajectory from medieval automata, such as Ismail al-Jazari's mechanical devices in 1206 and Leonardo da Vinci's robot knight around 1495, to modern neural networks, perceptrons (1957), and deep learning (1965). This volume explores AI's intersections with mythology, computing, medicine, and popular culture, including legends of ancient golems and Victorian steampunk machines, thereby providing a comprehensive illustrated timeline that connects early mechanical ingenuity to breakthroughs like AlphaGo's 2016 victory.

Publications and Creative Works

Non-Fiction Books

Clifford A. Pickover has authored over 50 non-fiction books, many of which explore the intersections of mathematics, science, art, and philosophy, and have been translated into more than a dozen languages, including Japanese, Chinese, Korean, French, Italian, German, Spanish, Portuguese, Polish, Greek, Turkish, Russian, Serbian, and Romanian. His non-fiction works include: Pickover's debut book, Computers, Pattern, Chaos and Beauty: Graphics from an Unseen World (St. Martin's Press, 1990), published at the height of early public fascination with chaos theory and fractals, pioneered the use of computer graphics to make complex mathematical concepts accessible to broad audiences, blending rigorous science with stunning visual art. The book explores hidden patterns in nature, music, genetics, and chaos; introduces techniques for visualizing data through fractals, cellular automata, biomorphs, and dynamical systems; and includes over 275 computer-generated images, many in color, to demonstrate 'a new way of seeing' the unseen world. It received high praise from leading figures, including Martin Gardner in Scientific American, who called it a 'truly stunning survey' of chaos and fractals' consequences, urging no informed layperson, artist, scientist, or mathematician to miss it, likening its revelations to those of telescopes or microscopes; Ian Stewart, who described it as 'A spectacular encounter between the art of the mathematician and the mathematics of art'; Ivars Peterson in Science News, who praised it as 'A cornucopia of visual ideas... at the frontiers of scientific and mathematical visualization'; and others, including Paul Hoffman and Ben Bacon, who highlighted its stimulating odyssey uniting art and science. Selected as one of the best science books of 1990 by Library Journal, this work launched Pickover's influential career emphasizing computers, creativity, lateral thinking, and visualization—topics that influenced later popular science and computational art. It remains notable for bridging disciplines during the emerging era of personal computing and fractal exploration, earning a reprint edition by Dover Publications in 2001. His early works, originating in the 1980s, laid the foundation for his prolific output, with titles like Computers and the Imagination (1991, St. Martin's Press) introducing readers to the creative potential of computing through explorations of patterns, chaos, and digital art. This book, building on his IBM research in visual computing, emphasized how computers could generate imaginative visuals and mathematical curiosities, influencing subsequent popular science writing. Pickover's non-fiction often groups into thematic clusters, beginning with mathematics and science in the 1990s. Books such as (1992, St. Martin's Press) delved into puzzles, fractals, and computational creativity, presenting mind-bending problems alongside computer-generated imagery to make abstract concepts tangible. Later entries like Keys to Infinity (1995, Wiley) examined infinite series, paradoxes, and geometric wonders, using accessible narratives to unpack profound ideas from number theory and topology. In the realm of scientific visualization, works including Frontiers of Scientific Visualization (1994, Wiley, co-edited with Stuart K. Tewksbury) highlighted techniques for rendering complex data, from fluid dynamics to biological structures, though Pickover's authored contributions focused on innovative imaging methods. Cultural and philosophical explorations appeared in titles like Dreaming the Future: The Fantastic Story of Prediction (2001, Prometheus Books), which chronicled methods from ancient oracles to modern futurism, blending history with speculative inquiry. Other notable volumes, such as Wonders of Numbers (2001, Oxford University Press) and (2012, Sterling), showcased his signature style of curating historical milestones with vivid illustrations and anecdotes. In recent years, Pickover has updated and expanded his catalog, with revisions reflecting contemporary advancements. Artificial Intelligence: An Illustrated History, From Medieval Robots to Neural Networks (2024, second edition, Union Square & Co.) traces AI's evolution through over 100 entries, incorporating recent developments in machine learning and ethics, including coverage of technologies such as DALL-E and ChatGPT. Revised editions published in 2025 include : From Pythagoras to the 57th Dimension, 250 Milestones in the History of Mathematics (Union Square & Co.), adding entries on modern topics like quantum computing, and The Physics Book: From the Big Bang to Quantum Resurrection, 250 Milestones in the History of Physics (Union Square & Co.), which incorporates discoveries such as the Higgs boson and gravitational waves. Pickover's books have achieved bestseller status in science and mathematics categories, earning praise for their engaging, illustrated format that democratizes complex subjects for broad audiences. (2009 original) won the 2011 Neumann Prize from the British Society for the History of Mathematics for its outstanding contribution to popularizing mathematical history. Reviews in outlets like Forbes commended the accessibility of works such as The Physics Book (2011 original), noting their conversational tone and visual appeal without compromising factual rigor. However, some critiques highlight a trade-off between this breadth and depth, suggesting the books prioritize inspirational overviews over exhaustive technical analysis, appealing more to enthusiasts than specialists.

Science Fiction: Neoreality Series

Clifford A. Pickover's Neoreality series introduces a distinctive style of speculative fiction known as "neoreality," which grounds narratives in established scientific principles while incorporating dream-like, surreal twists to explore alternate realities. This approach creates worlds separated from everyday existence by thin, imperceptible veils, blending rigorous concepts from physics, mathematics, and biology with hallucinatory elements to provoke reflections on perception and existence. The core of the series comprises four interconnected novels published in 2002 by The Lighthouse Press: Liquid Earth, The Lobotomy Club, Sushi Never Sleeps, and Egg Drop Soup. Additional science fiction works include Spider Legs (with Piers Anthony), The Heaven Virus, and Jews in Hyperspace. These works can be read in any order and feature protagonists encountering bizarre phenomena in familiar settings, such as a New York electronics store or the streets of New Orleans. For instance, The Lobotomy Club delves into surgically altered brains and quests for transcendence, while Liquid Earth depicts reality melting into fluid, psychedelic landscapes influenced by hyperspace physics. Central themes in the Neoreality series include parallel universes, fractal geometries, cosmic wormholes, and mind-expanding adventures that fuse science with surrealism, often featuring elements like intelligent spiders, hallucinating androids, and fractal-based intimacy. These narratives emphasize psychedelic explorations of consciousness and the boundaries of reality, echoing Pickover's non-fiction interests in mathematics and visualization without directly replicating factual expositions. Pickover extended the neoreality style into later works, such as Jews in Hyperspace (2009), which examines parallel dimensions near Jerusalem, intertwining religion, miracles, advanced technology, and physics in a tale of disappearing Orthodox Jews evaporating into higher realms. Similarly, (2007) incorporates themes of evolutionary destiny and inner space through a blend of humor, psychedelia, and speculative spiritual technologies, hinting at AI-like entities in hallucinatory contexts. These evolutions maintain the series' focus on surreal science while adapting to contemporary motifs like dimensional ethics and consciousness expansion.

Edited Collections and Columns

Clifford A. Pickover has edited several volumes that compile contributions from various experts, focusing on interdisciplinary themes at the intersection of science, art, and technology. His edited collections include:
  • Visions of the Future
  • Frontiers of Scientific Visualization
  • Future Health
  • The Pattern Book
  • Visualizing Biological Information
  • Fractal Horizons
  • Chaos and Fractals
His early edited collections include Spiral Symmetry (1992, World Scientific), which gathers essays on symmetrical patterns in nature and mathematics; Visions of the Future (1993, St. Martin's Press), exploring speculative advancements in science and society; and Frontiers of Scientific Visualization (1994, Wiley), featuring techniques for rendering complex data in visual forms. These works emphasize collaborative insights into emerging fields, drawing from physicists, artists, and computer scientists to illustrate conceptual innovations. In the mid-1990s, Pickover continued this editorial role with Future Health: Computers & Medicine in the 21st Century (1995, St. Martin's Press), compiling perspectives on computational applications in healthcare; The Pattern Book: Fractals, Art, and Nature (1995, World Scientific), which anthologizes discussions on fractal geometry's aesthetic and natural occurrences; and Visualizing Biological Information (1995, World Scientific), addressing graphical representations of genetic and molecular data. Later volumes such as Fractal Horizons: The Future Use of Fractals (1996, St. Martin's Press) and Chaos and Fractals: A Computer Graphical Journey (1998, Elsevier) further curate expert analyses on chaotic systems and their applications in modeling unpredictable phenomena. These collections have been noted for bridging academic research with broader accessibility, influencing discussions on computational creativity. Throughout his career, Pickover has contributed regularly to periodicals, most notably as the lead writer for the "Brain Boggler" column in Discover magazine from the 1990s onward, where he presented over 100 puzzles integrating mathematics, art, and science to engage general audiences. He also authored the "Brain-Strain" column for Odyssey magazine, extending similar mind-expanding challenges for younger readers. These columns, appearing monthly on magazine back pages, have fostered public interest in interdisciplinary thinking, with many puzzles later adapted for online platforms to reach wider audiences. Pickover's edited works and columns have significantly promoted public engagement with science, earning praise for inspiring creativity; as the New York Times noted, he "contemplates realms beyond our known reality," while the Los Angeles Times highlighted his role in "stretching the limits of computers, art, and thought."

Digital and Online Projects

WikiDumper.org

WikiDumper.org is a digital archive launched by Clifford A. Pickover on November 4, 2006, designed to preserve snapshots of Wikipedia articles nominated for deletion or otherwise rejected from the English Wikipedia. The project serves as an "appreciation page" for content deemed unworthy of inclusion in the main encyclopedia, capturing entries that range from the whimsical to the obscure before they are removed. By hosting these rejects, the site highlights the dynamic and selective nature of collaborative knowledge-building on Wikipedia. The blog's last post was in November 2013. Key features of WikiDumper.org include a blog-style repository of preserved articles, often presented with their original Wikipedia sources and licensing details under the GNU Free Documentation License. It functions as a searchable collection—leveraging Blogger's built-in search tools—featuring humorous, controversial, or niche topics such as unconventional chess variants, cultural trivia, and speculative scientific concepts. Accompanying each entry is minimal but pointed commentary from Pickover, emphasizing the value of diverse ideas in knowledge curation and questioning the criteria for what constitutes "encyclopedic" content. The site accepts user submissions, allowing contributors to nominate potential rejects for archival, though it does not outline strict inclusion standards. This approach underscores Pickover's motivation to critique the gatekeeping processes in digital encyclopedias, preserving material that might otherwise vanish into the ether of online edits. As of 2025, WikiDumper.org maintains a cultural footprint as a niche reference for discussions on information preservation and Wikipedia's editorial evolution, with early media coverage noting its role alongside broader archiving efforts like the Internet Archive. Described as a "popular blog" in contemporaneous reports, it has influenced conversations on digital ephemera without formal usage statistics publicly available, though its entries continue to be cited in explorations of online knowledge dynamics. No direct integrations with social media platforms are evident, but the project's ethos aligns with Pickover's wider online endeavors in fostering creative and intellectual discourse.

Recent Digital Initiatives

In the 2020s, Clifford A. Pickover has maintained and updated his personal website, pickover.com, featuring interactive mathematical applets such as visualizations of the Mandelbrot Set to engage visitors with fractal geometry and computational explorations. Pickover's social media presence on X (formerly Twitter) has grown significantly during this period, reaching over 180,000 followers by 2024 and surpassing 197,000 by July 2025, and over 208,000 as of November 2025, where he frequently shares mathematical puzzles, excerpts from his books, and commentary on science and creativity to foster public interest in these topics. These efforts complement his broader digital archiving work, including ties to tools like WikiDumper for preserving online content.

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