Dimensionality in Flatland

Considered as a literary classic, Flatland has been read and re-read in terms of its satirical value on rigid Victorian class structures, while at the same time, the component of inter-dimensional travel is often forgotten. Even today, more focuses are directed towards the historicist interpretation of Victorian values rather than the mathematical reasoning, despite the latter being the intrinsic conceptual framework for the book. Though dimensionality contributes to the satire, its interpretations are certainly not limited to it; in fact, these interpretations evolve with the progression of time. This essay will briefly identify its evolving significance in three time periods in particular: the Victorian era, early twentieth century, and today.

Beyond its satirical value, Edwin A. Abbott’s investigation into dimensionality can also be considered as his way of reconciling religious visions with science. During the Victorian era, the general devotion to Christianity was stronger than ever, yet the biblical doctrine was not enough to suffice people who now seek reason behind natural phenomena.[1]Natural Theology, published in 1802, was a product of such pursuit. In the book, William Paley suggests that natural phenomena are the result of God’s creation, supplementing to the context of the Bible.[2] However such theological explanation was later confronted by the publication of Origin of Species in 1859, which shifted the use of nature from supporting genesis to supporting evolution.[3] These two seemingly contradictory natural theories gave the Victorians an imperative task in reconciling science and religion, and from there derived the interest in the study of dimensionality.[4] Abbott was among one of many scholars who showed such curiosity. For Abbott, his interest started with his belief in the commonness of spiritual vision.[5] As an academic as well as a theologist, he also believes that there is a logical explanation behind such vision. The dimensionality in Flatland is where he executed his investigation. In the book, the visit of the Sphere to Flatland was perceived by the protagonist, A. Square, as a line that stretches and compresses, appears and disappears,[6] a behavior that parallels the behavior of spiritual vision. However, with the mathematical setup, the logic behind such perception is indisputable. Towards the end of the book, the protagonist also suggests the possibilities of a “Thoughtland” and the possibilities of its creature presenting itself in the three-dimensional world in the same ethereal matter the Sphere presents in Flatland.[7] With the use of mathematical induction, Abbott scientifically and logically presents the possibility of a higher being suggested by religion. And this attempt in using dimensionality to understand religion was as much as Abbott’s interest as a product of the Victorian culture.

Coming into the 20th century, the dimensionality in Flatland, together with the new definition of the fourth dimension, opened up the discourse of science, which set the way for our understanding of the field today. After the publication of Albert Einstein’s Special Theory of Relativity in 1906, where he presented the idea of time-space as the fourth dimension, Flatland’s conception of the higher dimension was re-examined. As science was newly acknowledged as an academic discipline, it welcomed both ideas,[8] defining Einstein’s conception as space-temporal, and Flatland’s conception as strictly spatial. Attempts in bringing the two ideas into conversation were also made. Sphereland by Dionys Burger is only one example. The book acts as a sequel to Flatland, adopting the two-dimensional plane of Abbott’s, but with a clear reference to Einstein’s concept of the higher dimension. In the book, A. Square discovers that his land is not flat after all, rather, it is curved into the surface of a sphere, a direct reference to the Einstein visual representation of the space-time continuum.[9] Though there are still no concrete algorithms in both or either theory of the fourth dimension, it is the accepting attitude and the dialectical approach to the problem that represents the openness of science in the 20th century, which set the way for our excitement about the possibilities within the field.

Today, in an age where the enthusiasm for understanding dimensionality is higher than ever, Abbott’s mathematical approach still plays an integral part. With the new technological tools now accessible, referencing upon the bases of Flatland, the 21st-century people are able to take the interpretation of dimensionality into all types of directions, disciplines, and communication formats. In 2001, Ian Stewart published his fiction Flatterland as the newest sequel to Abbott’s flatland. In his book, through the encounters of the protagonist Victoria Lines, A. Square’s great-great-granddaughter, and other characters that are abstractions of scientific ideas, such as Schrodinger’s cat, Stewart examines our present understanding of shape, time, dimension, and the universe.[10] Beyond literature into the space that is unique to now, the social media space, there is information about dimensionality constantly being exchanged. Countless videos about the idea of higher dimensions are posted on YouTube. “Visualizing 4D Geometry” posted by The Lazy Engineer is only one of those who uses Flatland’s plane as a reference point in digitally rendering a four-dimensional apparition.[11]

If the investigation into dimensionality means the reconciliation of religion of science in the Victorian era, a representation of the openness of science in the 20th century, in the age of globalization and interconnectivity, it is then interesting to consider what it means now. What metaphorical meaning can the fourth dimension now take on? Where would it lead us in searching for a globalized identity? These and many other questions are the ones waiting to be answered.

[1] Richard Evans, “The Victorians: Religion and Science, Gresham College,” 2011, https://www.gresham.ac.uk/lectures-and-events/the-victorians-religion-and-science.

[2] Charlotte Barrett, “Science and Religion,” Great Writers Inspire, accessed February 27, 2021, https://writersinspire.org/content/science-religion.

[3] Ibid.

[4] Thomas F. Banchoff, “From Flatland to Hypergraphics: Interacting with Higher Dimensions,” Department of Mathematics, Brown University, accessed February 26, 2021, https://www.math.brown.edu/tbanchof/abbott/Flatland/ISR/.

[5] Christopher G.White, Other Worlds: Spirituality and the Search for Invisible Dimensions, (Cambridge: Harvard University Press, 2018), ProQuest Ebook Central, 35.

[6] Edwin A. Abbott, Flatland: A Romance of Many Dimensions, (Project Gutenberg, 2008), kindle, loc 1009.

[7] Ibid, loc 1230.

[8] “History & Philosophy of Science,” American Association for the Advancement of Science, accessed February 27, 2021, https://www.aaas.org/programs/dialogue-science-ethics-and-religion/history-philosophy-science.

[9] Thomas F. Banchoff, “Flatland: A New Introduction,” accessed February 28, 2021, https://www.math.brown.edu/tbanchof/abbott/Flatland/Publications/intros/banchoff.pdf.

[10] “Flatterland: Like Flatland Only More So by Ian Stewart,” Goodreads (Goodreads, April 18, 2002), https://www.goodreads.com/book/show/17033.Flatterland.

[11] “Visualizing 4D Geometry – A Journey Into the 4th Dimension [Part 2],” YouTube, posted by “The Lazy Engineer,” Aug 12, 2017, https://youtu.be/4URVJ3D8e8k.