Recently, I have been playing with the whimsical idea of a fictional number, one that everyone has somehow missed when counting from four to five.
The idea itself is not new. Gird and Bleem (between 3 and 4), Sorf (between 1 and 2) as well as many others (such as SCP-033) have explored this concept somewhat, but I was considering it specifically in the context of a type of insanity that could be inflicted upon a player in a table-top RPG, and as an interesting mathematical and etymological puzzle.The basic concept is that, if you, the player, were to be afflicted with this specific insanity, you would become convinced that there is a number between four and five that is somehow subconsciously ignored by everyone— except you. You can perceive it, or are at least aware of it. In this context, the player would likely have the opportunity to come up with their own terminology to try to convey the concept to others in-character, but for this thought experiment, I've assumed that there is some sort of historical record of the missing number. In this way then, we can imagine a fictional history of its application, and create some life-like etymology around it.
For the purposes of this thought experiment, the name we arrived at for the missing number is torve. We were looking for a moniker that had a verisimilitude with other "number words" without aping them completely. We needed something that was distinct, but felt like it belonged in a list of numbers, that is to say. It also needed to sound fundamentally English (this is my main criticism of some of the other imagined numbers, such as bleem). Torve was derived¹ from Proto-Germanic *trabō, meaning "fringe," cognate to Greek δρεπᾰ́νη/drepánē, meaning "scythe" from which we can derive drepne as a greek translation for torve for constructing geometric terms.
In order to fit sensibly into base ten, we would assume the existence of a whole set of torve-related numbers: torfteen, twenty-torve, torty, torty-torve, one-point-torve, one-over-torve, negative-torve, etc.²
However, applying even basic mathematics or reasoning to any of these numbers seems to result in logical conflicts/paradoxes (which would explain why the orderly universe simply skips over it). For instance, torve seems to be either even or odd, depending on which direction you're counting from. Even simply multiplying or dividing torve causes trouble. If two times four is eight, and two times five is ten, then what is two times torve? Nine? ³
It may seem on the surface that some of this could be mitigated by imagining additional missed numbers. For instance, let us imagine another new number called enevre hidden between nine and ten. Five plus five still equals ten, five plus four still equals nine, so five plus torve equals enevre. It follows then that torve plus torve is equal to nine (this means that nine is now somehow an even number). So then, what is enevre divided by two? We would assume torve-point-five, which seems consistent, but this begs the question: if half of nine is now torve, what is two times four-point-five? It has to be a whole number, but it cannot be eight, nine, or enevre.
For the same reasons, applying division at any stage creates paradoxes as well, a symptom of torve being simultaneously even and odd.. Half of torve can neither be two, nor two-point-five. If torve is even, it must have an integer half, necessitating a whole number existing paradoxically between two and two-point-five. It is assumed that the rest of mathematics should continue to function as expected, but there is no way to rectify this with torve. If two times three is six, then six minus two should be equal to two times two. But two times two is four, and yet six minus two is torve.
If we take torve into the realm of geometry, there are some interesting constructions that we could imagine. You could draw a shape with torve sides: a drepnagon— or perhaps we could call it a drepnus in the fashion of "rhombus." You could presumably also draw a star-polygon with torve points: a drepnagram. You could then move up a dimension⁵ and construct a new platonic solid out of ten drepnagons⁴, with three meeting at each vertex, a drepnic decahedron. Assuming this solid follows the same rules as other regular polyhedra, it would have a corresponding dual solid, swapping the number of vertices with the number of faces. We can call this solid a drepkaidecahedron, constructed from torfteen triangles (torve times three plus one), torve meeting at each vertex.⁶
In a game-mechanic context, torve allows for some interesting results (if we hand-wave the mathematical paradoxes). For instance, our insane character would perceive themselves to have a torfth finger on each hand. It is impossible for them to determine which are the fingers on their hands that they hadn't noticed all this time prior to becoming insane, but it somehow permits an additional ring-slot. Each hand has an extra finger, and yet somehow the total number of fingers on both hands still equals ten.
The only way to rectify the paradoxical nature of torve with the surrounding orderly reality is to treat it as a fictionalizing agent. Functionally, this means that attempting to apply torvic mathematics to any real system quickly results in the system/construction progressively de-reïfying, becoming increasingly imaginary as a result of losing compatibility with any logical context. Gameplay-wise, this means that a character's insanity would permit, depending on the situation, acts that seem to cross or blur the line between reality and fiction. If they and their comrades are fighting five enemies, there may be an additional foe that only the insane character may perceive under certain circumstances. Yet, any money taken from the defeated foe will be completely real— as will injuries resulting from the fight.⁷
All dice used by the player stricken with insanity would necessarily have an additional torfth face. It would be assumed that part of the character's insanity makes it very difficult for them to make sense of numbers in general. So, as an interesting game mechanic, the player may add or subtract one from any dice-roll (determined by coin flip) a certain number of times per day.⁸ 20's roll over to 1's and visa-versa. 4's and 5's respectively may become torves.⁹ These results will appear non-anomalous to sane observers, but with strange, inconsistent results that increase the fictivity of a situation (which may have other unexpected results). Hitting an enemy with attacks of a damage value equal to torve or it's derivatives several times (or, ideally, torve times) will make the enemy significantly more imaginary.
As a closing thought, I suppose one could characterize the weirdness that arrives from the introduction of torve to a system is that it is essentially attempting to convert base ten into base eleven without actually changing the values of the existing digits. It doesn't work, and it doesn't even make any sense, but that is the point, I suppose. Even trying to create a consistent logical system around the concept seems intractable, but it makes for an interesting thought experiment, trying to nail down the impossible boundaries of a fundamentally illogical system.
¹ presumably originating from the Proto-Indo-European *der- (“to tear, pluck”), or *dʰrebʰ- (“to beat, crush, make or become thick”). Other sources suggest a slavic origin, but we are assuming it to be a native P.I.E. word for our purposes. The real English derivatives from this stem are possibly drub (“to beat”), or Middle English trappe (“trappings, personal belongings”).
² What shall we call the set of numbers between one quadrillion and one quintillion? This necessitates extending the etymology to Latin as well. The actual Latin cognate for this stem is drappus, meaning "piece of cloth," but following the evolution of other number words in Latin, we might imagine a form like drappe. Following the pattern, we would call our new power of ten one drappillion, however, the way we define this number ordinally is somewhat confusing. If one trillion is equal to ten to the power of twelve, it would follow that one quadrillion is ten to the power of [12 + 3], which would normally be fifteen, but is now torfteen. This would mean that torfteen would replace fifteen as a multiple of three, which is nonsensical. So, one drappillion should be equal to ten to the power of eighteen, ten to the power of seventeen, or ten to the power of fifteen, depending on which direction you're counting from, and which starting point.
³ I do not wish to speculate on what sort of headaches taking a number to the torfth power would create.
⁴ I arrived at ten for the number of faces by assuming that the torvic decahedron follows a similar pattern of faceting rules as the cube and the dodecahedron, lying intermediate between the two. If we picture a cube, and begin at a single face, then we can move outwards from that face, adding a new face at each edge, and allowing those faces to connect to one another as well, forming a bowl. At this point, we have [x + 1] facets, where x is the number of sides a facet-shape has. On a cube, the dihedral angle is equal 90°, and thus there is only room for one more face, sealing off the bowl ([x + 1] + 1, thus 6). On a dodecahedron, there is room for a whole additional bowl ([x + 1] + [x + 1], thus 12). Intermediate between these two solutions would be a ring of x facets meeting at a point. Logically, this is inconsistent, as we have already established that a ring of x facets surrounds a single face; the only real shape that can do both is a triangle. But if we ignore this inconsistency, we have a form with [x + 1] + x facets, or [2x + 1] facets. Assuming we are not inventing additional fictional numbers to solve metanumeral paradoxes, we could reason that two times torve is equal to nine, resulting in ten facets total for a torvic decahedron. This also feels appropriate, as the pattern of platonic solids and their number of faces seems to have a gap between the octahedron and the dodecahedron. The one remaining problem with this solution is that we have already established that three drepnagons meet at each vertex, and yet we have just described torve faces meeting at the vertex opposite the base, meaning that while the calculated number of sides fits a sort of logic if you squint your eyes, the result is as nonsensical as the number itself.
⁵ Upon further consideration, I think that 10 sides is not the complete answer. Reconsiddering the geometry, it seems like it would most likely follow that one would start with two bases, extend them out to form a bowl, a ring of torve drepnagons each. These bowls clasp one another, forming the complete torvic decahedron. However, it is only a decahedron when counted from two opposing ends, making the two halves that are added together. If instead, one counts faces starting at one opposing face, and progresses to the opposing face, one will find that it is a torvic hendecahedron.
⁶ Moving to the fourth dimension, one could speculate on additional polychora made out of torve-based polyhedra, or perhaps a "torfplex" made from an torvic number of tetrahedra. More interesting to me, however, is the notion of moving a dimension higher still, and attempting to wrap ones head around what a torfth dimension of space would entail, of torve-polytopes, or if geometry and logic apply at all within an imaginary orientation of space.
⁷ Continuing this line of reasoning, we could imagine a class of secondary solids stemming from these. For instance, there would be a whole new class of Johnson Solids based around the drepnagon, such as a drepnagonal pyramid, or a drepnagonal prism. We could also imagine a new set of Kepler–Poinsot polyhedra, star-polyhedra based around the drepnic decahedron and drepkaidecahedron, though I fear that attempting to wrap my head around what these might be like is somewhat beyond me. Because the nature of symmetry is impossible to pin down when considering a shape that has simultaneously an even or odd number of sides, it is difficult to say what other solids could be formed from combinations of drepnagons and other regular polygons, though at the very least we might imagine there being a set of truncated solids.
⁷ We would imagine that sane observers of this situation would claim to have seen the insane character come about the money/injuries/etc. via ordinary means— from the insane perspective, this is their befuddled minds filling in the gaps left by their inability to perceive the number torve.
⁸ Perhaps torve times per day.
⁹ 14's and 15's as well on d20's may become torfteens; d%'s are treated as two separate dice rolls
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