Tolkienian geometry or the point of Tom Bombadil

J.R.R. Tolkien gave a tantalising clue to the role of dimensions within his work through the title of his first successful book, “The Hobbit or There and Back Again”.

The Line

The alternative title to The Hobbit alludes to the linear nature of Bilbo’s adventure: he travels from his comfortable Hobbit Hole to the lair of Smaug, helps the Dwarves recover their lost heirlooms while gaining riches and The One Ring for himself as well, before returning back to his dwelling. In very simple terms, he goes there and back again.

The story arc of Lord of the Rings follows a similar pattern in that Frodo  and his hobbit companions also go there and back again, beginning and ending their epic journey in the Shire. Tolkien also signals the drawing to a close of a greater story arc, namely that of the Silmarillion, in the eventual continuance of Frodo’s journey, accompanying Gandalf and the Elves to the Grey Havens. This final stage in Lord of the Rings is the completion of Galadriel’s journey of there (Middle Earth) and back again (Valinor), finishing the saga of the Silmarillion.

Its worth noting that traversing forwards and backwards along a straight line transcribes a wave when moving in a direction perpendicular to that line.

Waves or cycles formed by going there and back again on X or Y axes. Spiral formation going there and back again along both X and Y axes simultaneously.

If we think of this perpendicular motion (represented by the Z axis above) as time then the tales of The Hobbit and The Silmarillion/Lord of the Rings form cycles both in the narrative and geometrical sense. Weapons may also give a sense for Tolkien’s view on the ‘dimensionality’ of a particular race. Using the designations found in Lord of the Rings, Men and Hobbits are mainly associated with swords, indicating the line as the generative form of the narrative cycle for these races.

The Circle

The circle, as seen above, is generated by going there and back again simultaneously along two lines or axes set perpendicular to each other. This circular motion transcribes a spiral when moving in a direction perpendicular to the plane of the circle.

The circle is most notably identified by the rings in Lord of the Rings. They denote the ability of characters associated with them to operate along an extra dimension to those who ‘walk the line’. Elrond, Galadriel and Gandalf are all characters with this extra dimension while the Nazgûl are enslaved by their rings within the “unseen world” or wraith world. We conclude that men are not capable of being able to properly comprehend the unseen world while certain sources note the great power held by the High Elves within this realm. The tales of these multidimensional characters weave in and out of the linear narrative of Lord of the Rings though spirals, like cycles, eventually return to their original point, which in the case of these characters is The Grey Havens and The West. Glimpses of the unseen world experienced by the bearers of the One Ring may explain why they accompany the host into The West – their story line has now become a circle.

Sauron is certainly also associated with the circle, being the creator of the master ring. Tolkien possibly hints at a deeper understanding possessed by Sauron of the powers of the ring in his illustrations for the cover of the Lord of the Rings.

Original illustration by J.R.R. Tolkien for the Book Cover of The Fellowship of the Ring

The central motif of Sauron’s Eye was adopted by early editions of the book. Examining the geometrical composition of the Eye, a point on a central, vertical line is encompassed by a surrounding circle which is itself enclosed in the ellipse of the Eye. The two dimensions discussed so far are present in its construction and the outer ellipse can be interpreted to signify Sauron as the master of these dimensions as a circle is a member of the family of ellipses: Sauron can warp the dimensions of the circle in all manner of ways, unknown even to the Elves. As a side note, an interesting element of the illustration is that the three Elven rings possess a jewel but the One Ring has none; Tolkien’s drawing suggests Narya, the Ring of Fire, is the jewel of the One Ring.


Having identified the circle with the elves, the bow as representative of the arc of a circle is a natural choice as the favoured weapon of the elves. Through the favoured axe of the dwarves we  can now ascertain the awareness of this delving race. The axe is in the form of a ‘T’ (or ‘t’ for a single-bladed axe)  which suggests an incomplete expansion into the second dimension or “unseen world”. The decoration on Gimli’s helmet in the films suggests a triangle, a first crude step from a line towards the “perfected” circle.


The Sphere

The sphere is a circle of circles and hence represents a great leap in terms of dimensionality within Tolkien’s universe.

There are few references to spheres within Tolkien’s work with the most  known example being the Palantíri or Seeing-stones. These stones were used in communication within Middle Earth and beyond, requiring users of great strength of will and wisdom. The inherent dangers in using these devices was demonstrated in the madness of Denethor and the catatonic state induced in Pippin after gazing into the Orthanc Palantír – even the One Ring did not induce such adverse reactions in its users. Both Sauron and Sauruman possessed and could operate the spheres but, according to Gandalf, did not have the skill to create them. Indeed it is told Fëanor, greatest of all Elven smiths, was their creator and only he had such skill among the Eldar. One cannot help but speculate Tolkien’s vision of the powers and dangers inherent in the dawning Age of Communication are instilled within the Palantíri .

Another reference to the sphere is given by Tolkien in an interview from the 1960s which can be viewed below. At around six minutes into the interview, he outlines his obsession with Atlantis and his realisation of the mythical island in Númenor, the island gifted to Men for their aid in the conquest of Melkor, the first dark lord.

The island allowed them to gaze on the Undying Lands but they were forbidden to set foot on its shores. In defiance, they attempted to sail to the Land of the Valar but Eru , the One God, caused the ships of the Númenoreans and their island to be sunk below the ocean. To prevent future incursions, Eru reshaped the flat world of Arda to be a sphere, forever placing the Undying Lands beyond the reach of Men. Interestingly, the elves can reach these lands by the Straight Road, a diminishing of their world to the one dimensional line hinted at by Galadriel when tested by the One Ring.

The Point

There are  two principle characters to whom this dimension applies, Tom Bombadil and his wife Goldberry. Tom has a strong affinity with trees and can be seen to be the anthropomorphism of a tree, a being rooted to the spot. Goldberry is also a point but, unlike Tom, is the point in motion. She is thus associated with the flowing of the river and the changes in the seasons. Together they are masters of their home and untouched by time and its shaping of the lives of all other beings. Indeed two points define a line which suggests Tom and Goldberry may well form the root for all paths of Men, Hobbits and Elves. Ents in their ponderous ways seen to span between the perspectives of point and line. Tolkien certainly had great affection for both Tom and trees, naming two small anthologies of his work, “Tree and Leaf” and “The Adventures of Tom Bombadil”,  after them. In leaving Tom and his wife out of the film adaptations of Lord of the Rings, did director Peter Jackson miss the point?