Ralph Dumain

Martin Gardner, Mathematical Games, & the Fourth Dimension

On this site

Dumain contributions

Mathematical Games: Counting Systems and the Relationship between Numbers and the Real World” by Martin Gardner, Scientific American, “Mathematical Games” column, June 1968 [Excerpt including acknowledgments of R. Dumain et al for solutions to Baker’s Solitaire problem, not included in the anthologies]

The fourth dimension

A New Facet of Cubism: ‘The Fourth Dimension’ and ‘Non-Euclidean Geometry’ Reinterpreted” by Linda Dalrymple Henderson (See also Henderson; Kern; below)

Alfred Jarry, Marcel Duchamp, ’Pataphysics, & the Fourth Dimension (excerpts) by Andrew Hugill

Black mystic in hyperspace by R. Dumain

Review: Sydney Padua, The Thrilling Adventures of Lovelace and Babbage by R. Dumain

Arithmophobia summarized by R. Dumain

Nonsense is the Fourth Dimension of Literature by Gelett Burgess

Gustav Spiller Against the Fourth Dimension

Frank Blighton & the Fourth Dimension

Into the Fourth Dimension” by Frank Blighton

Into the Fifth Dimension” by Frank Blighton

The Hop on Hyperspace” by Frank Blighton

Friedrich Engels on Empiricism, Spiritualism, Science, Mysticism, and Philosophical Naivete

La naturesplorado en la mondo de la spiritoj de Friedrich Engels, elgermanigita de Vilhelmo Lutermano (in Esperanto)


“Mathematical Games” by Martin Gardner (Scienterrific American, January 1977, p. 8);  parody by Danny Abelson & Ellis Weiner, National Lampoon, January 1977, p. 50.

Web guides & bibliographies

Board Games & Related Games & Recreations: Web Guide

Homage to Martin Gardner (October 21, 1914 – May 22, 2010) [all links incorporated into the present web page]

Bibliography of Board Games & Related Games & Recreations (As of 30 August 1994) in the collection of Ralph Dumain

Bibliography & Catalog of Board Games & Related Games & Recreations: Games Collection of Ralph Dumain: Supplement # 1 (1994-2008)

Alfred Jarry’s “How to Construct a Time Machine”: A Web Guide

Algorithmic Puzzles: Selected Bibliography

Lewis Carroll’s Logic Games, Mathematical Recreations, Puzzles & Paradoxes

Mathematical Fiction & Related Works: A Guide

Vladimir Nabokov: Science Fiction, Artificial Languages, Ars Combinatoria, Narrative Structure, Martin Gardner, Play: An Arbitrary Bibliography

Ramon Llull & His Influence: Select Bibliography & Web Guide

Doubt & Skepticism: A Directed Minimal Bibliography & Web Guide

Enrique Gaspar y Rimbau: El anacronópete — The First Time Machine

H. G. Wells’ The Time Machine: Selected Bibliography

Selected Non-fictions: Table of Contents by Jorge Luis Borges; Eliot Weinberger (ed.,tr.), Esther Allen (tr.), Suzanne Jill Levine (tr.)

Jorge Luis Borges: Selected Study Materials on the Web

The Cavalier: Covers & Contents

J. U. Giesy (John Ulrich, 1877-1948) & His Collaborators

Science Fiction & Utopia Research Resources: A Selective Work in Progress

On other sites

Dumain contributions


Martin Gardner @ Reason & Society

Testimonials (includes history of my engagement with Martin Gardner)

Martin Gardner Dead at 95

Testimonial 55: Ralph Dumain (Martin Gardner Testimonials)


Martin Gardner - Wikipedia, the free encyclopedia

List of Martin Gardner Mathematical Games columns - Wikipedia, the free encyclopedia

Martin Gardner bibliography - Wikipedia, the free encyclopedia

Essential web sites & bibliographical guides

Martin Gardner Home Site (Martin Gardner Centennial 1914-2014)

Martin Gardner Testimonials

Gathering for Gardner

Gardner Index by Carl W. Lee

Martin Gardner's Collections of "Mathematical Games" Columns by Brian Raiter

Martin Gardner: Mathematical Games

Martin Gardner's Mathematical Games: The Entire Collection of His Scientific American Columns (CD-ROM)

The New Martin Gardner Mathematical Library (Cambridge University Press)

Autobiography & documentary

A Quarter Century of Recreational Mathematics, by Martin Gardner, originally published in Scientific American,  August 1998; reprinted May 29, 2010

Martin Gardner Mathemagician,” episode of “The Nature of Things,” CBC, 1996 (YouTube, 46:04)

Miscellaneous tributes

Martin Gardner Communications Award, July 1994, with Gardner interview (YouTube, 14:08)

Game Inventor: Martin Gardner (a tribute from Kadon Enterprises, Inc.)

John Conway Reminiscences about Dr. Matrix and Bourbaki, by Colm Mulcahy, Scientific American blog, April 1, 2015.

Inspired by Martin Gardner

The Lives and Games of Martin Gardner - Numericana

Bulgarian Solitaire

Ask Dr. Matrix

The Mathemagician and the Pied Puzzler: A Collection in Tribute to Martin Gardner, edited by Elwyn Berlekamp & Tom Rodgers (Natick, MA: AK Peters, 1999)

The fourth dimension (hyperspace) & Flatland(s)

The Unexpected Hanging And Other Mathematical Diversions by Martin Gardner (1991 ed.) *

Martin Gardner | The Church of the Fourth Dimension by Giulio Prisco, October 13, 2012

Fourth dimension - Wikipedia, the free encyclopedia

Four-dimensional space - Wikipedia, the free encyclopedia

Fourth dimension in literature - Wikipedia, the free encyclopedia

Hyperspace - Wikipedia, the free encyclopedia

Johann Karl Friedrich Zöllner - Wikipedia, the free encyclopedia

Spiritualism - Wikipedia, the free encyclopedia

Gustav Spiller - Wikipedia, the free encyclopedia

Edwin Abbott Abbott - Wikipedia, the free encyclopedia

Flatland - Wikipedia, the free encyclopedia

Charles Howard Hinton - Wikipedia, the free encyclopedia

Sphereland - From Wikipedia, the free encyclopedia

Flatterland - From Wikipedia, the free encyclopedia

Rudy Rucker - From Wikipedia, the free encyclopedia

Flatland: A Romance of Many Dimensions, with Illustrations by the Author, A SQUARE [Edwin Abbott Abbott]

Flatland 1965 [film: YouTube]

Flatland: The Film (2007) [YouTube]

Flatland (2007 film) - Wikipedia, the free encyclopedia

The Flatland Store (Flatland: The Movie)

Flatland: The Movie - Wikipedia, the free encyclopedia (2007)

Flatland 2: Sphereland (2012) [IMDB]

Flatland — more YouTube video & audio

C4D | The Church Of The Fourth Dimension

Llull, Bruno, Borges, and the fourth dimension

Borges on Hinton

Charles Hinton and His Cubes by David Auerbach

Deflating Hyperspace” by David Pacchioli

Natural Science and the Spirit World in Dialectics of Nature by Friedrich Engels

Blindness (1977) by Jorge Luis Borges (mentions Spiller)

Mathematical Fiction: Motif=Higher/Lower Dimensions: Search here

Film & television

4D Man (1959) (full movie, YouTube, 1:25:00)

The Borderland - Wikipedia, the free encyclopedia (episode of The Outer Limits, aired 16 December 1963,

The Borderland (full episode, YouTube, 51:22)


Musée Patamécanique.
Note the sculpture Time Machine by Dr. Ezekiel Borges Plateau.
The tour concludes with “The Laboratory for the Study of Advanced Fourth Dimensional Mechanics.”
Daren Elsa NiBelly, Vice-Curator, “also teaches Patamechanics at the Center for Advanced Fourth Dimensional Inquiry at Tlon University.”

Assorted references

Abbott, Edwin Abbott. The Annotated Flatland: A Romance of Many Dimensions; introduction and notes by Ian Stewart. Cambridge, MA: Perseus Publishing, 2002. (Re-published: New York: Basic Books, 2008.)

Antliff, Mark. “The Fourth Dimension and Futurism: A Politicized Space,” The Art Bulletin, Vol. 82, No. 4 (Dec., 2000), pp. 720-733.

Arithmophobia: An Anthology of Mathematical Horror, edited by Robert Lewis. Polymath Press, 2024. See:

“Splinters” by Miguel Fliguer & Mike Slater, pp. 22-46.
“Manifold Thoughts” by Patrick Freivald, pp. 47-59.
“A Presence Beyond the Shadows” by David Lee Summers, pp. 245-258.

See also: Arithmophobia summarized by R. Dumain.

Bentley, Nancy. “The Fourth Dimension: Kinlessness and African American Narrative,” Critical Inquiry, Vol. 35, No. 2, Winter 2009, pp. 270-292.

See also Du Bois, below.

Blacklock, Mark. The Emergence of the Fourth Dimension: Higher Spatial Thinking in the Fin de Siècle. Oxford, UK: Oxford University Press, 2018.

A Bouquet for the Gardener: Martin Gardner Remembered, by Mark Burstein et al. New York: The Lewis Carroll Society of North America, 2011.

Dewdney, A. K. The Planiverse: Computer Contact with a Two-Dimensional World. New York: Springer (Copernicus imprint), 2000. (1st ed. 1984)

Du Bois, W. E. B. “A Vacation Unique,” appendix to Dark Voices: W. E. B. Du Bois and American Thought, 1888–1903 by Shamoon Zamir (Chicago; London: The University of Chicago Press, 1995), p. 217-225; and see Zamir’s discussion of the text. See also Bentley, above.

Gardner, Martin. Logic Machines and Diagrams. New York: McGraw-Hill, 1958. (2nd. ed., University of Chicago Press, 1982.)

Gardner, Martin. “The Puzzles in Ulysses,” Semiotica, vol. 57, nos. 3-4 (1985), pp. 317-330. Also in Gardner, The Night is Large: Collected Essays, 1938-1995 (New York: St. Martin’s Press, 1996), pp. 354-367.

Gardner, Martin. Undiluted Hocus-Pocus: the Autobiography of Martin Gardner. Princeton: Princeton University Press, 2013.

* Gardner, Martin. The Unexpected Hanging And Other Mathematical Diversions. 2nd ed. Chicago: University of Chicago Press, 1991. With updated chapters & bibliography, and an afterword. The original edition: New York: Simon & Schuster, 1969.

“The Church Of The Fourth Dimension,” 1969 ed., Chapter 6, pp.65-75; 1991 ed., same +  pp. 246-247 (in Afterword) + Bibliography, p. 256.

“An adventure in hyperspace at the Church of the Fourth Dimension,” “Mathematical Games” column, January 1962.

“Flatlands,” 1969 ed., Chapter 12, pp. 136-146; 1991 ed., same + p. 249 (in Afterword) + Bibliography, p. 259.

“Fiction about life in two dimensions,” “Mathematical Games” column, July 1962.

See also: “The pleasures of doing science and technology in the planiverse” [Dewdney], “Mathematical Games” column, July 1980.

And see also other columns & chapters:

“Is it possible to visualize a four-dimensional figure?”, November 1966.
“The orders of infinity, the topological nature of dimension and "supertasks",” March 1971.
“Hypercubes,” in Mathematical Carnival (1975)
“Spheres and Hyperspheres,” in Mathematical Circus (1979)

Note: I read The Unexpected Hanging in 1969, my likely introduction to Flatland and the fourth dimension.

Granville, William Anthony. The Fourth Dimension and the Bible. Boston: Richard G. Badger, 1922.

The Fourth Dimension and the Bible (1922) by Sam Dolbear (The Public Domain Review).

Henderson, Linda Dalrymple. The Fourth Dimension and Non-Euclidean Geometry in Modern Art. Revised ed. Cambridge, MA: The MIT Press, 2013. (1st ed.: Princeton: Princeton University Press, 1983.)

Henderson, Linda Dalrymple. “Mysticism as the ‘Tie That Binds’: The Case of Edward Carpenter and Modernism,” Art Journal, Vol. 46, No. 1 (Mysticism and Occultism in Modern Art), pp. 29-37. See also:

Editor’s Statement: Mysticism and Occultism in Modern Art,” Art Journal, Vol. 46, No. 1, pp. 5-8.

Henderson, Linda Dalrymple. “Mysticism, Romanticism, and the Fourth Dimension,” in The Spiritual in Art: Abstract Painting 1890- 1985 [exhibition catalog. Los Angeles County Museum of Art, November 23, 1986 - March 8, 1987] (New York: Abbeville Press, 1986), pp. 219-37.

Huneker, James. “The Disenchanted Symphony,” in Melomaniacs (New York: Charles Scribner's Sons, 1902), pp. 324-346.

Kern, Stephen. The Culture of Time and Space, 1880–1918, with a new preface. Cambridge, MA: Harvard University Press, 2003. First edition, 1983:

1. The Nature of Time
2. The Past
3. The Present
4. The Future
5. Speed
6. The Nature of Space
7. Form
8. Distance
9. Direction
10. Temporality of the July Crisis
11. The Cubist War

In 1913 Apollinaire commented that Cubists have followed scientists beyond the third dimension and “have been led quite naturally . . . to preoccupy themselves with new possibilities of spatial measurement which, in the language of the modern studies, are designated by the term: the fourth dimension.” 31 There was a popular interest in the fourth dimension in France at that time, which might have inspired the Cubists. 32 [p. 145]

31. Guillaume Apollinaire, Cubist Painters (1913; rpt. New York, 1944), 13. [See pp. 13-17, 61, 98, 129, 135 — RD]

32. E. Jouffret, Traité élémentaire de géometrie à quatre dimensions (Paris, 1903), 153. This connection was made by Linda Dalrymple Henderson in an article on possible influences of physics and geometry on the Cubists: “A New Facet of Cubism: ‘The Fourth Dimension’ and ‘Non-Euclidean Geometry’ Reinterpreted,” The Art Quarterly (Winter 1971): 411-433. She refutes the facile connections between Cubism and science made by Paul M. Laporte and others and shows that although the Cubists could not have known about Einstein’s relativity or Minkowski’s space-time theory, they might have learned about the fourth dimension and non-Euclidean geometry from their friend, the insurance actuary Maurice Princet. Although Picasso denied having discussed the fourth dimension with Princet, she speculates that it is possible that he or the other Cubists picked up a suggestion of it from him indirectly. [p. 338]

Pawlowski, Gaston de. Voyage au pays de la quatrième dimension. Paris: Bibliothéque-Charpentier, 1912.

Journey to the Land of the Fourth Dimension, translated by Brian M. Stableford. Encino, CA: Black Coat Press, 2009. [Based on the 1923 expanded French edition. See the translator’s introduction via this link.]

Pickover, Clifford A. Jews in Hyperspace. Kindle Edition, 2009.

Pickover, Clifford A. Surfing through Hyperspace: Understanding Higher Universes in Six Easy Lessons. New York: Oxford University Press, 1999.

Robbin, Tony. Shadows of Reality: the Fourth Dimension in Relativity, Cubism, and Modern Thought. New Haven: Yale University Press, 2006.

Throesch, Elizabeth. Before Einstein: The Fourth Dimension in Fin-de-Siècle Literature and Culture. London: Anthem Press, 2017. (Open access by chapter)

Throesch, Elizabeth. “Nonsense is the Fourth Dimension of Literature: Hyperspace Philosophy, the ‘New’ Mathematics, and the Alice Books,” in Alice beyond Wonderland: Essays for the Twenty-First Century, edited by Cristopher Hollingsworth, foreword by Karoline Leach (Iowa City: University of Iowa Press, 2009), pp. 37-52.

See also: Nonsense is the Fourth Dimension of Literature by Gelett Burgess

Van Heijenoort, Jean. “Friedrich Engels and mathematics” (1948), in Selected Essays (Napoli: Bibliopolis; Atlantic Highlands, NJ: Distributed in the U.S.A. by Humanities Press, 1985), pp. 123-151. References for article & whole book, pp. 153-166.

White, Christopher. “Seeing Things: Science, the Fourth Dimension, and Modern Enchantment,” The American Historical Review, Vol. 119, No. 5 (December 2014), pp. 1466-1491.

On the religious crisis & occultist tendency of the time, as experienced by Charles Howard Hinton, on his promotion of the 4th dimension & considerable influence, & on the controversies surrounding the potential of the mathematical imagination.

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