Non-Euclidean Literature

Non-Euclidean Literature

Through Lovecraft, Daumal, Hermes Trismegistus, and more: Why it is important to change your perspective​

Of all the different forms of art, literature is the medium which generates the most imaginative response. With painting and architecture, visualisation is a finished product, supplying viewers with impressions based on what they see. With writing on the other hand visualisation is conjured from the mind of the reader, directly. Their deductions are based on perceptions that don’t exist in reality, and even could not exist. Of course, capacity to imagine the unreal or impossible has also been explored by visual artists. Take M. C. Escher. In a single drawing (called “Relativity”) flights of stairs extend in all directions, people sit upside-down, doorways lie on their sides. The picture plane confronts our sense of physical reality and our assumptions of what constitutes the realm of possibility, by forcing us to perceive a reality that doesn’t make physical sense.

But seeing a visual representation of the physically unimaginable isn’t the same as imagining the unimaginable itself, which is what literature is empowered to do. For example, take the city of R’lyeh, an ancient ruin in the middle of the Pacific Ocean first mentioned in The Call of Cthulhu by the writer HP Lovecraft. In the story, one character is described as having “climbed interminably along the grotesque stone moulding—that is, one would call it climbing if the thing was not after all horizontal.” Later a character is “swallowed up by an angle of masonry which shouldn’t have been there; an angle which was acute, but behaved as if it were obtuse.” Recalling his impressions of the sunken city, a character reflects “the geometry of the dream-place he saw was abnormal, non-Euclidean, and loathsomely redolent of spheres and dimensions apart from ours.”

The term “non-Euclidean” used here is worth exploring. The immediate association is naturally to Euclidean geometry, which deals with linear objects and shapes. Non-Euclidean geometry, on the other hand refers to the geometry of curved surfaces: a geometry which challenges ordinary perceptions. Think of a triangle with three sides at angles of ninety degrees. Impossible, right? But imagine standing anywhere on the equator. Walk in a straight line until you reach the North Pole, turn ninety degrees, and walk again until you hit the equator, turn ninety degrees once again in the same direction, start walking, and you will end up back where you started. A perfect triangle with three right angles.

Important to non-Euclidean mathematics is the fact that space is curved in four dimensions. Accordingly, mathematicians and physicists acknowledge that, in real world calculation problems, Euclidean geometry does not work On this basis, the idea of the non-Euclidean has been co-opted by analogy to refer to any physical (or metaphysical) entity or place which exceeds traditional human understanding. For Lovecraft this means above all the Old Ones or the Outer Gods, entities said to dwell outside of thought and beyond time itself. A similar thought is involved in the gnostic figure of the demiurge, in which the creator of the known universe is conceived as the secondary products of a more fundamental and distant creation.

Myths and stories have always been used as methods of transmitting cultural knowledge. Accordingly, most stories are allegorical, that is, presenting a symbolic structure in the service of a central message. Probably the non-Euclidean literary work which goes the furthest down this path is Mount Analogue by René Daumal. First published in France in 1952 and translated into English seven years later, the novel’s full title is Mount Analogue: A Novel of Symbolically Authentic Non-Euclidean Adventures in Mountain Climbing. The plot of the novel takes shape as an expedition to a mysterious mountain whose location cannot be perceived directly, but only when a traveller realizes that he has travelled further than would otherwise be possible in a straight line. An amalgamation of other mythical and physical mountains —Sinai, Nebo, Olympus, and Everest—Mount Analogue is understood to connect Heaven and Earth, and as such “its summit must be inaccessible, but its base accessible to human beings as nature made them.” Or as Daumal puts it, “the door to the invisible must be visible.”

Ultimately Mount Analogue is a tale about initiation—an adventure theoretically available to anyone, but not necessarily to everyone. Ironically Daumal died before the novel could be completed with his characters still at base camp. Perhaps this fact itself presents an allegory of the futility of striving for perfection—that joy and meaning is found most often in the journey itself rather than the destination. Similar metaphysical perspectives have been presented before throughout literature. In Dante’s Paradiso, concluding a journey from another mountain—Mount Purgatory—Dante states that we may never understand the transmundane: “ma non eran da ciò le proprie penne,” or roughly “but my own wings were too weak for that.” In the prologue to Nietzsche’s Thus Spake Zarathustra, the title character descends from a mountain to rejoin the world: “I am weary of my wisdom, like the bee who has gathered too much honey.”

The word “apotheosis” is also worth considering here. Colloquially, apotheosis means an apex experience, whether figuratively or literally, like the summit of a mountain. But technically apotheosis refers to an elevation to divinity. But how is this achieved? In his notes accompanying Mount Analogue, Daumal reflects that: “You cannot always stay on the summits. You have to come down again… so what’s the point? Only this: “What is above knows what is below, what is below does not know what is above.” The formulation echoes almost perfectly the famous statement from the Emerald Tablet, a central text in Western esotericism attributed to Hermes Trismegistus: “That which is above is like to that which is below, and that which is below is like to that which is above.” The idea is that the physical and more generally the terrestrial sphere reflects a higher metaphysical dimension. As Emanuel Swedenborg put it, in the form of the idea of correspondences, and spiritual analogies: “Just as a knowledge of the Principles of Geometry enables man to measure distant suns and their movements, while seated in his observatory, so a knowledge of the Principles of Correspondence enables man to reason intelligently from the Known to the Unknown.”

Probably the best-known work by a living writer involving non-Euclidean themes is House of Leaves by Mark Z Danielewski, published in 2000. The book associates a set of stories, narratives, reviews and academic literature around something called The Navidson Record, a documentary film chronicling what happens after a man discovers that a hallway in his house is a quarter of an inch longer on the inside than on the outside. As time progresses, while the exterior of the house remains the same, the hallway, and the house’s interior, become much larger, with enormous rooms and chambers branching off in strange directions. The eventual result is a labyrinthine structure that the characters, and the reader, struggle to follow. Indeed, like the mythical Theseus in the labyrinth finding his way through by following a ball of thread tied to the entrance, one character uses a tied fishing line to trace his progress. And as with the Theseus myth, here, too, is a minotaur.

House of Leaves is a mammoth creation; the full-colour edition is 709 pages long, with footnotes on most pages. Some footnotes even have footnotes themselves, referring to and quoting countless texts, both real and imaginary, across multiple languages. Typographically, the book is also a maze, changing physics in the same way as the house; footnotes appear upside-down, parts of the text is mirrored and placed at certain points on the page, one section is even written in braille.

Echoing philosopher Jacques Derrida’s work on deconstruction, especially his experimental book Glas, House of Leaves invites the reader to form their own meaning by making the reader a character in the story, finding (or not finding) their own way through the labyrinth. What a reader finds meaningful, and meaningless, in the book depends entirely on their perspective at the time of reading. Read it again years later and it will tell an entirely different story.

In all these literary tales, the key variable upon which all enjoyment depends is whether the reader is willing to change their perspective. Sometimes, the reader is actually forced to look at things differently. Perhaps it is important to remember that our perspective can always be changed. The Delphic oracle called Socrates the wisest man in Athens, and subsequently the world, because he knew that nobody really knew anything. The more complex things get, the simpler they become.

As above, so below.

Adam James Pollock is a writer living in Ireland.

Adam James Pollock is a writer living in Ireland.

Of all the different forms of art, literature is the medium which generates the most imaginative response. With painting and architecture, visualisation is a finished product, supplying viewers with impressions based on what they see. With writing on the other hand visualisation is conjured from the mind of the reader, directly. Their deductions are based on perceptions that don’t exist in reality, and even could not exist. Of course, capacity to imagine the unreal or impossible has also been explored by visual artists. Take M. C. Escher. In a single drawing (called “Relativity”) flights of stairs extend in all directions, people sit upside-down, doorways lie on their sides. The picture plane confronts our sense of physical reality and our assumptions of what constitutes the realm of possibility, by forcing us to perceive a reality that doesn’t make physical sense.

But seeing a visual representation of the physically unimaginable isn’t the same as imagining the unimaginable itself, which is what literature is empowered to do. For example, take the city of R’lyeh, an ancient ruin in the middle of the Pacific Ocean first mentioned in The Call of Cthulhu by the writer HP Lovecraft. In the story, one character is described as having “climbed interminably along the grotesque stone moulding—that is, one would call it climbing if the thing was not after all horizontal.” Later a character is “swallowed up by an angle of masonry which shouldn’t have been there; an angle which was acute, but behaved as if it were obtuse.” Recalling his impressions of the sunken city, a character reflects “the geometry of the dream-place he saw was abnormal, non-Euclidean, and loathsomely redolent of spheres and dimensions apart from ours.”

The term “non-Euclidean” used here is worth exploring. The immediate association is naturally to Euclidean geometry, which deals with linear objects and shapes. Non-Euclidean geometry, on the other hand refers to the geometry of curved surfaces: a geometry which challenges ordinary perceptions. Think of a triangle with three sides at angles of ninety degrees. Impossible, right? But imagine standing anywhere on the equator. Walk in a straight line until you reach the North Pole, turn ninety degrees, and walk again until you hit the equator, turn ninety degrees once again in the same direction, start walking, and you will end up back where you started. A perfect triangle with three right angles.

Important to non-Euclidean mathematics is the fact that space is curved in four dimensions. Accordingly, mathematicians and physicists acknowledge that, in real world calculation problems, Euclidean geometry does not work On this basis, the idea of the non-Euclidean has been co-opted by analogy to refer to any physical (or metaphysical) entity or place which exceeds traditional human understanding. For Lovecraft this means above all the Old Ones or the Outer Gods, entities said to dwell outside of thought and beyond time itself. A similar thought is involved in the gnostic figure of the demiurge, in which the creator of the known universe is conceived as the secondary products of a more fundamental and distant creation.

Myths and stories have always been used as methods of transmitting cultural knowledge. Accordingly, most stories are allegorical, that is, presenting a symbolic structure in the service of a central message. Probably the non-Euclidean literary work which goes the furthest down this path is Mount Analogue by René Daumal. First published in France in 1952 and translated into English seven years later, the novel’s full title is Mount Analogue: A Novel of Symbolically Authentic Non-Euclidean Adventures in Mountain Climbing. The plot of the novel takes shape as an expedition to a mysterious mountain whose location cannot be perceived directly, but only when a traveller realizes that he has travelled further than would otherwise be possible in a straight line. An amalgamation of other mythical and physical mountains —Sinai, Nebo, Olympus, and Everest—Mount Analogue is understood to connect Heaven and Earth, and as such “its summit must be inaccessible, but its base accessible to human beings as nature made them.” Or as Daumal puts it, “the door to the invisible must be visible.”

Ultimately Mount Analogue is a tale about initiation—an adventure theoretically available to anyone, but not necessarily to everyone. Ironically Daumal died before the novel could be completed with his characters still at base camp. Perhaps this fact itself presents an allegory of the futility of striving for perfection—that joy and meaning is found most often in the journey itself rather than the destination. Similar metaphysical perspectives have been presented before throughout literature. In Dante’s Paradiso, concluding a journey from another mountain—Mount Purgatory—Dante states that we may never understand the transmundane: “ma non eran da ciò le proprie penne,” or roughly “but my own wings were too weak for that.” In the prologue to Nietzsche’s Thus Spake Zarathustra, the title character descends from a mountain to rejoin the world: “I am weary of my wisdom, like the bee who has gathered too much honey.”

The word “apotheosis” is also worth considering here. Colloquially, apotheosis means an apex experience, whether figuratively or literally, like the summit of a mountain. But technically apotheosis refers to an elevation to divinity. But how is this achieved? In his notes accompanying Mount Analogue, Daumal reflects that: “You cannot always stay on the summits. You have to come down again… so what’s the point? Only this: “What is above knows what is below, what is below does not know what is above.” The formulation echoes almost perfectly the famous statement from the Emerald Tablet, a central text in Western esotericism attributed to Hermes Trismegistus: “That which is above is like to that which is below, and that which is below is like to that which is above.” The idea is that the physical and more generally the terrestrial sphere reflects a higher metaphysical dimension. As Emanuel Swedenborg put it, in the form of the idea of correspondences, and spiritual analogies: “Just as a knowledge of the Principles of Geometry enables man to measure distant suns and their movements, while seated in his observatory, so a knowledge of the Principles of Correspondence enables man to reason intelligently from the Known to the Unknown.”

Probably the best-known work by a living writer involving non-Euclidean themes is House of Leaves by Mark Z Danielewski, published in 2000. The book associates a set of stories, narratives, reviews and academic literature around something called The Navidson Record, a documentary film chronicling what happens after a man discovers that a hallway in his house is a quarter of an inch longer on the inside than on the outside. As time progresses, while the exterior of the house remains the same, the hallway, and the house’s interior, become much larger, with enormous rooms and chambers branching off in strange directions. The eventual result is a labyrinthine structure that the characters, and the reader, struggle to follow. Indeed, like the mythical Theseus in the labyrinth finding his way through by following a ball of thread tied to the entrance, one character uses a tied fishing line to trace his progress. And as with the Theseus myth, here, too, is a minotaur.

House of Leaves is a mammoth creation; the full-colour edition is 709 pages long, with footnotes on most pages. Some footnotes even have footnotes themselves, referring to and quoting countless texts, both real and imaginary, across multiple languages. Typographically, the book is also a maze, changing physics in the same way as the house; footnotes appear upside-down, parts of the text is mirrored and placed at certain points on the page, one section is even written in braille.

Echoing philosopher Jacques Derrida’s work on deconstruction, especially his experimental book Glas, House of Leaves invites the reader to form their own meaning by making the reader a character in the story, finding (or not finding) their own way through the labyrinth. What a reader finds meaningful, and meaningless, in the book depends entirely on their perspective at the time of reading. Read it again years later and it will tell an entirely different story.

In all these literary tales, the key variable upon which all enjoyment depends is whether the reader is willing to change their perspective. Sometimes, the reader is actually forced to look at things differently. Perhaps it is important to remember that our perspective can always be changed. The Delphic oracle called Socrates the wisest man in Athens, and subsequently the world, because he knew that nobody really knew anything. The more complex things get, the simpler they become.

As above, so below.

Through Lovecraft, Daumal, Hermes Trismegistus, and more: Why it is important to change your perspective​
Through Lovecraft, Daumal, Hermes Trismegistus, and more: Why it is important to change your perspective

Of all the different forms of art, literature is the medium which generates the most imaginative response. With painting and architecture, visualisation is a finished product, supplying viewers with impressions based on what they see. With writing on the other hand visualisation is conjured from the mind of the reader, directly. Their deductions are based on perceptions that don’t exist in reality, and even could not exist. Of course, capacity to imagine the unreal or impossible has also been explored by visual artists. Take M. C. Escher. In a single drawing (called “Relativity”) flights of stairs extend in all directions, people sit upside-down, doorways lie on their sides. The picture plane confronts our sense of physical reality and our assumptions of what constitutes the realm of possibility, by forcing us to perceive a reality that doesn’t make physical sense.

But seeing a visual representation of the physically unimaginable isn’t the same as imagining the unimaginable itself, which is what literature is empowered to do. For example, take the city of R’lyeh, an ancient ruin in the middle of the Pacific Ocean first mentioned in The Call of Cthulhu by the writer HP Lovecraft. In the story, one character is described as having “climbed interminably along the grotesque stone moulding—that is, one would call it climbing if the thing was not after all horizontal.” Later a character is “swallowed up by an angle of masonry which shouldn’t have been there; an angle which was acute, but behaved as if it were obtuse.” Recalling his impressions of the sunken city, a character reflects “the geometry of the dream-place he saw was abnormal, non-Euclidean, and loathsomely redolent of spheres and dimensions apart from ours.”

The term “non-Euclidean” used here is worth exploring. The immediate association is naturally to Euclidean geometry, which deals with linear objects and shapes. Non-Euclidean geometry, on the other hand refers to the geometry of curved surfaces: a geometry which challenges ordinary perceptions. Think of a triangle with three sides at angles of ninety degrees. Impossible, right? But imagine standing anywhere on the equator. Walk in a straight line until you reach the North Pole, turn ninety degrees, and walk again until you hit the equator, turn ninety degrees once again in the same direction, start walking, and you will end up back where you started. A perfect triangle with three right angles.

Important to non-Euclidean mathematics is the fact that space is curved in four dimensions. Accordingly, mathematicians and physicists acknowledge that, in real world calculation problems, Euclidean geometry does not work On this basis, the idea of the non-Euclidean has been co-opted by analogy to refer to any physical (or metaphysical) entity or place which exceeds traditional human understanding. For Lovecraft this means above all the Old Ones or the Outer Gods, entities said to dwell outside of thought and beyond time itself. A similar thought is involved in the gnostic figure of the demiurge, in which the creator of the known universe is conceived as the secondary products of a more fundamental and distant creation.

Myths and stories have always been used as methods of transmitting cultural knowledge. Accordingly, most stories are allegorical, that is, presenting a symbolic structure in the service of a central message. Probably the non-Euclidean literary work which goes the furthest down this path is Mount Analogue by René Daumal. First published in France in 1952 and translated into English seven years later, the novel’s full title is Mount Analogue: A Novel of Symbolically Authentic Non-Euclidean Adventures in Mountain Climbing. The plot of the novel takes shape as an expedition to a mysterious mountain whose location cannot be perceived directly, but only when a traveller realizes that he has travelled further than would otherwise be possible in a straight line. An amalgamation of other mythical and physical mountains —Sinai, Nebo, Olympus, and Everest—Mount Analogue is understood to connect Heaven and Earth, and as such “its summit must be inaccessible, but its base accessible to human beings as nature made them.” Or as Daumal puts it, “the door to the invisible must be visible.”

Ultimately Mount Analogue is a tale about initiation—an adventure theoretically available to anyone, but not necessarily to everyone. Ironically Daumal died before the novel could be completed with his characters still at base camp. Perhaps this fact itself presents an allegory of the futility of striving for perfection—that joy and meaning is found most often in the journey itself rather than the destination. Similar metaphysical perspectives have been presented before throughout literature. In Dante’s Paradiso, concluding a journey from another mountain—Mount Purgatory—Dante states that we may never understand the transmundane: “ma non eran da ciò le proprie penne,” or roughly “but my own wings were too weak for that.” In the prologue to Nietzsche’s Thus Spake Zarathustra, the title character descends from a mountain to rejoin the world: “I am weary of my wisdom, like the bee who has gathered too much honey.”

The word “apotheosis” is also worth considering here. Colloquially, apotheosis means an apex experience, whether figuratively or literally, like the summit of a mountain. But technically apotheosis refers to an elevation to divinity. But how is this achieved? In his notes accompanying Mount Analogue, Daumal reflects that: “You cannot always stay on the summits. You have to come down again… so what’s the point? Only this: “What is above knows what is below, what is below does not know what is above.” The formulation echoes almost perfectly the famous statement from the Emerald Tablet, a central text in Western esotericism attributed to Hermes Trismegistus: “That which is above is like to that which is below, and that which is below is like to that which is above.” The idea is that the physical and more generally the terrestrial sphere reflects a higher metaphysical dimension. As Emanuel Swedenborg put it, in the form of the idea of correspondences, and spiritual analogies: “Just as a knowledge of the Principles of Geometry enables man to measure distant suns and their movements, while seated in his observatory, so a knowledge of the Principles of Correspondence enables man to reason intelligently from the Known to the Unknown.”

Probably the best-known work by a living writer involving non-Euclidean themes is House of Leaves by Mark Z Danielewski, published in 2000. The book associates a set of stories, narratives, reviews and academic literature around something called The Navidson Record, a documentary film chronicling what happens after a man discovers that a hallway in his house is a quarter of an inch longer on the inside than on the outside. As time progresses, while the exterior of the house remains the same, the hallway, and the house’s interior, become much larger, with enormous rooms and chambers branching off in strange directions. The eventual result is a labyrinthine structure that the characters, and the reader, struggle to follow. Indeed, like the mythical Theseus in the labyrinth finding his way through by following a ball of thread tied to the entrance, one character uses a tied fishing line to trace his progress. And as with the Theseus myth, here, too, is a minotaur.

House of Leaves is a mammoth creation; the full-colour edition is 709 pages long, with footnotes on most pages. Some footnotes even have footnotes themselves, referring to and quoting countless texts, both real and imaginary, across multiple languages. Typographically, the book is also a maze, changing physics in the same way as the house; footnotes appear upside-down, parts of the text is mirrored and placed at certain points on the page, one section is even written in braille.

Echoing philosopher Jacques Derrida’s work on deconstruction, especially his experimental book Glas, House of Leaves invites the reader to form their own meaning by making the reader a character in the story, finding (or not finding) their own way through the labyrinth. What a reader finds meaningful, and meaningless, in the book depends entirely on their perspective at the time of reading. Read it again years later and it will tell an entirely different story.

In all these literary tales, the key variable upon which all enjoyment depends is whether the reader is willing to change their perspective. Sometimes, the reader is actually forced to look at things differently. Perhaps it is important to remember that our perspective can always be changed. The Delphic oracle called Socrates the wisest man in Athens, and subsequently the world, because he knew that nobody really knew anything. The more complex things get, the simpler they become.

As above, so below.

Adam James Pollock is a writer living in Ireland.

"Relativity", by M.C. Escher

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