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The Mathematicians
The Mathematicians
More information can be found in the Sect Write-up

From their burg of Radian, in the geometric centre of Mechanus (so they reckon), the Mathematicians do their number-crunching and calculating. There's an entrance to the Labyrinthine Portal here, and apart from the modrons, there cutters are the best-qualified to operate the barmy thing. The more adventurous members of the sect chase the results of their calculations and explore the distant parts of Mechanus in search of ... well, whatever it is these basher seek.

Radian is also home to a large colony of moignos, both rational and irrational. These bashers live in the burg's central clearing, the Set Square

Sectols in the Mathematicians are not called "Sectols". They are referred to as the "Prime Digit", or often just the "Digit". The Digit of the Mathematicians is selected by the previous Digit. Thus, before Rikard died, he chose one of his most intelligent pupils to take his place, and then that one chose his favourite, and so on. The current Prime Digit of the Mathematicians is Laretuous Quinton. Laretuous has been the Digit for about eight years now, and he's the youngest Sector to date, aged only thirty-one years.

The three principal sectors (the equivalent to factors of the factions) of the sect are brother and sister pair of tiefling bashers named Sine and Cosine and a rogue modron called Tangent. These names are the traditional ones adopted by Mathematician sectors; the real names of the cutters are kept a secret by the sect until they are replaced. Nobody really knows why, but the Mathematicians are a very traditional lot, prone to following reams of dogma at the slightest provocation.

The Institute of Digitology
(by Belarius)
Recently, the Mathematicians of Mechanus have discovered a type of magic quite different from the traditional method -- a type that involved great quantities of mathematics. The Digitologists of Zakhara (a region on the prime world of Toril, better known as the Land of Fate) uses extensive mathematical equations to very precisely draw the spell into effect, rather than bluntly drive a formula into one's head until memorised and then regurgitate it later. They develop an understanding of the magic which allows them to augment it further than was previously thought possible.

Of course, the Mathematicians celebrated this accomplishment. For years, they have been trying to find the reverse of Wild Magic, and Digitology seems to be it -- a rigorous and highly organised form of magic which is not only predictable, but can be exploited to augment the powers of magic even further. News of this discovery spread to Sigil, where it was received with virtually no interest: "Don't the Mathematicians use math for their magic anyway?" Only a few people really understood the potential of this type of magic. Before long, the Guvners (always looking for a good loophole) had gone through the right authorities to form a collaborative institute of research with the Mathematicians, to try to discover the limits of Digitology. This facility, creatively titled the Institute of Digitology, has become a bit of a hotspot.

Located near the Fortress of Disciplined Enlightenment (in planar terms), the Institute is made up of the most brilliant Guvner mages and various interested Mathematicians. To better understand the topic of their research, the Institute has recruited a fair number of wide-eyed Zakharan Primes, all Digitologists, to teach and learn at the Institute. Overall, the facility seems fairly unobtrusive, on the surface.

Investigators have found, however, that there are elements to the Institute which have made several other groups uncomfortable. Reliable information has reached the ears of the Athar that the Zakharan Digitologists believe it is a crime to attempt to spread the belief that no gods exist -- and have insisted that this rule be enforced in the Institute. A lawful member of the Athar was captured and put to death for casually speaking on the subject. Of course, the Athar are grievously offended by what they view as 'thought policing' and 'a breach of diplomatic decency.' The resulting friction between the Guvners and the Athar in Sigil is evident as Guvner-funded temples have received significantly more attention from the Athar (led by the antisocial Hobard).

Wild mages are also disturbed at what they see as a threat to their own style of magic. Though they are incapable of doing any direct harm to the Institute (since it is located in a Plane which negates wild magic), some have tried to intimidate the Guvners in Sigil with threats and a very small number have actually enlisted the aid of normal mages to commit acts of terrorism against the Institute.

One final concern is that the modrons will learn the secrets of the magic and become all the more powerful. After all, many high-ups are already spellcasters, and this will make them all the more powerful. Fortunately, since the modrons are rarely a threat to anyone outside of Mechanus, this is a minor concern at best.

Mathematics and Planewalking
(by Alex Roberts)

An essay on why mathematical study of the planes is relevant,
by Rigora Inflexion, a prime factor [high-up] of the Mathematicians

In my forays across the planes, I have discovered a number of interesting things:

The Spire on the Outlands: This spire is definitely defined by a mathematical equation. Trouble is, no-one can agree on what equation. (Of course, the Bleak Cabal claim that no such good sense exists, but I know what I've seen) Because space curves near the base of the spire, all attempts to measure its curvature have so far failed. Visual estimates from far off don't square with what's seen close to.

Infinity: Not even true-born planars know everything about infinity, even though it touches their lives so fundamentally. A prime called Georg Cantor seems to have had the dark of it quite well. He said that there was not just one kind of infinity, but at least two. He talked about 'countable' and 'uncountable' infinities.

Hold on, I hear you cry, how can you count infinity? Well, berk, it assumes you've got infinite time. But if you stumble on something uncountable, you couldn't count it even with infinite time. The classical example is of the whole numbers, or integers as they are called by the Mathematicians and Guvners. The integers are the definitive countable set. You can go 'One, two, three...' forever, and you'll never miss one if you've got long enough. But if you try to include all real numbers, including the so-called irrationals, like pi (bane of modrons!), you'll find that there's no way to account for all of them. You'll always miss infinitely many for every and any one you count.

To see it, just consider how many numbers differ from an infinitely long number like pi by just a single digit. Get the idea? Now, the question for the infinitude of planes is, which things in the planes are countable, and which uncountable? It is reasonably transparent that people are countable, and arbitrary points uncountable. But when it comes to layers of the Abyss, no-one's sure. If they are uncountable, then some must be empty. What's in an empty layer?

Space dimensions: Here's a topic that invariably confuses those not used to operating in strange environments. Many primes maintain that space is three-dimensional, and that time is a fourth dimension. It emerges that this is a pitifully inadequate explanation when you travel the planes. For example, all planes with a discernible form and more than one layer (except Bytopia, but I'll return to that) have at least four space dimensions: An up-down dimension, a forward/backward dimension, a left/right dimension and an up layer/down layer dimension.

There are exceptions. Mechanus has only one layer and therefore three space dimensions can describe it adequately. Bytopia has two layers, but they meet in a plane. This is like separating numbers into positive and negative. You still only have one line of numbers, but some are positive and others negative. Likewise, half of Bytopia is Shurrock and the other half is Dothion. In this way, Bytopia has two layers but only three space dimensions. Gehenna has four layers, but is a three-dimensional space and all the layers are finite.

The inner planes are always five-dimensional; for example, the plane of Fire has the dimensions up/down, left/right, forward/backward, magma/smoke and radiance/ash. The Astral Plane, just to be awkward, has no layers and no time, but is in fact a bizarre non-space. This leads to many bizarre results.

Limbo and the Abyss are so badly defined that their dimensionality is not a comprehensible concept. It has been suggested that there exist planes we can never visit that are home to two-dimensional beings who would treat a visit to the planes we experience much as a clueless prime experiences the Astral for the first time, especially if they were bodily adrift in the Void.

Speaking of the prime, it might be useful to point out that the prime may have a layered structure too. The prime most people think of is one of perhaps three main primes (one in which magic works but there is no phlogiston, one we know about already, and one from which I am writing this which has no inherent magic at all), and in addition to these, there are a number of 'pocket primes', realms resembling the primes we have mentioned, but actually more like demi-planes in size and inhabited by over-powers. At least, that's the theory.

A Study in Grey
(by Randir)

A group of Mathematician from Mechanus have begun a project unlike any which have gone before. Their goal is to prove the existence of the Least Common Denominator, the moral factor which is common to all moralities. If they find such a proof, they have theorised that all moralities would shift towards grey. A common ground could be found, encouraging discourse and promoting the development of Unified Morality.

They need not find this LCD, for its very existence would shake the very rock on which so many fundamentalists base their faith.

Recently, the Mathematicians have heard of a ancient sect located in the Outlands, and supposedly the Hinterlands. Although these reports are merely rumours, this sect is believed to be the "proof of the LCD" finding them would be to find the see on the planes abstract concepts are physical reality.

The Unificationists
A Segment of the Mathematicians
(by Randir)

"e pluribus unum" | "from many one"

The unificationists believe that the entire multiverse is a process, who's sole purpose is to become god. In the beginning, the multiverse was a single thing, imperfect through ignorance. At some point outside of time, this ignorance promoted its division into many parts. Godhead was corrupted and shattered into millions of pieces.

From these shards of consciousness came the powers, for originally all men were gods. However, as the Shattering took hold, godhead was corrupted by the flesh, and all of the moralities of the multiverse were unleashed.

It is hence the goal of all to reform the perfect whole, this time without the inexperience, and ignorance which plagued it so long ago.

DESCRIPTION: Unificationists all wear a single shade of gray, and speak slowly in a single tone. All Unificationists look exactly the same, sound exactly the same, and smell exactly the same. they are forbidden from having names, and will not use them. They all cover their eyes with dark glasses, and paint their lips a dull black.

SPECIAL BENEFITS: Unificationists can find common ground with all other sects and factions. Even the Blood War is very important to the Reformation. As such they automatically receive a favorable reaction from anyone, even fiends.

SPECIAL HINDRANCES: Unificationists may not obstruct any activities which promote the resolution of problems. And may not perform any act which encourages diversity.

A Character Kit
(by James Neal)

The Fraternity of Order believes in reductionism on an abstract, theoretical level. The Mathematicians of Mechanus attempt to apply reductionist theory in actual practice to the infinite complexity of Mechanus' clockworks. The beliefs of the Mathemusicians go one step beyond reductionism - perhaps translationism would be the word for it? According to the Mathemusicians, if everything can ultimately be reduced to such atomics as numbers and rules, then everything can, when seen in enough detail, be interpreted as anything else. The pattern, trajectory and speed of the flight of a flock of birds, could reduced the numbers describing that phenomenon, and those numbers could be translated into a mathematical equation.

The complex, programmed sequences of a mechanical clockwork construct (even, in theory, one of infinite size) can be reduced to numbers which can be translated into musical notation. The Mathemusicians believe that since everything can be reduced to abstract primitives, it is their task to translate everything they can into the pure, abstract, and beautiful of abstractions - music. They are currently obsessed with recording and comprehending the entire mechanical program sequence of Mechanus' gears, much like the Fraternity of Order and the Mathematicians, only their ultimate end intended for this information is to translate it into one immense, sprawling masterpiece of bewildering complexity, insanely elaborate counterpoint, and melody so long-range as to sound utterly aims to human comprehension.

ABILITY REQUIREMENTS: Intelligence 17, Dexterity 12


NON-WEAPON PROFICIENCIES: Bonus - Artistic Expression (Composing), Musical Instrument (player's choice); Recommended - Language (Modron), Planar Survival (Mechanus), Engineering

SPECIAL NOTE: The Composition NWP of a Mathemusician is based on their Intelligence, not their Wisdom.

SKILL MODIFICATIONS: -10 to CW and PP; +5 to RL; +15 to DN

EQUIPMENT: As per normal for a bard.

SPECIAL BENEFITS: Like their companions, the Mathematicians, their keen minds, devotion to the study of patterns, and day-to-day living on Mechanus has given them an intimate familiarity with the Labyrinthine Portal. They can calculate, in 1d3 turns, the correct path to any location within Mechanus that they have either been to or have had described to them in intimate detail.

Their music is also, when attended to closely enough, extremely captivating, at least to certain kinds of people. Anyone who begins to pay attention to a Mathemusician's playing must roll over (not under!) their Intelligence on a d20, or become so enraptured in its subtleties that they lose all track of the world around them and become lost in a world of musical abstraction. The DM may wish to assign bonuses or penalties to the roll according to the musical tastes of the target.

SPECIAL HINDRANCES: Because their musical compositions are oriented more as expressions of their philosophy than as works of art intended to capture the human heart, they lack the normal bardic ability to sway the reactions of crowds, and to inspire their allies in battle.

NOTE: Mathemusicians have the same abilities to learn wizard magic, and to counter music-based magic with their playing, that other bards have; they also abide by the same armour and weapons restrictions. However, unlike other bards, they may be LG or LE in addition to the four neutral alignments. Mathemusicians may not join a faction, as they are considered to be part of the Mathematician sect.

Set Theory
A Mathematician Speaks of his Theory of Everything
(by Randir)

Now, I have published a couple of papers on the notions of set theory to define the sum total of everything, and I think that is a great starting point for explaining Mathematician philosophy.

The rortiest thing is that the SET BASED UNIVERSE is highly ARATIONAL. In this it is particularly applicable to the Outer Planes. The fundamental notion that "Existence is the Set of all Sets" implies that:

"Anything that can be thought of does in fact exist" and

to a different degree...

"Anything which can't be thought of does in fact exist" and

in a very weird sense...

"Nothing does indeed exist..."

This all makes a lot of problems for logicians, philosophers, and other mixed nuts... Filled with logical contradictions (which largely exist in spite of themselves) and apparent paradoxes...

Oddly, all statements are then true and false at the same time each in greater and lesser measure. However, many neat ideas such as equality and singulars are even more important, since they form a new logical basis.

Some things will never change!

More online information can be found in the
Sect Write-up
For more "official" details on this sect please see page 12 of TSR's Planes of Law: Mechanus book.

Copyright 1998 by Jon Winter,
gearwork art and symbol by Belarius, moigno by Jeremiah Golden

Consult the Mimir Again