A/N: This is based on the premise that Bill's dimension was Flatland - a two-dimensional, strictly-regulated world inhabited by geometric figures, where each one's place in society is determined by the number of their sides. Flatland is narrated through the memoirs of A Square, who is shown the existence of a tridimensional world - only to be imprisoned by authorities to keep him from speaking of it. In the context of this story, said memoirs are exactly how Bill will find out about other dimensions, too.
The parts written in italics are lifted straight from Flatland, by Edwin A. Abbott.

"You think those chains are tight? Imagine living in the Second Dimension. Flat minds in a flat world with flat dreams. I liberated my dimension!"
– Bill Cipher, Gravity Falls

"Prometheus up in Spaceland was bound for bringing down fire for mortals, but I–poor Flatland Prometheus–lie here in prison for bringing down nothing to my countrymen. Yet I exist in the hope that these memoirs, in some manner, I know not how, may find their way to the minds of humanity in Some Dimension, and may stir up a race of rebels who shall refuse to be confined to limited Dimensionality."
– A Square, Flatland

Our Soldiers and Lowest Class of Workmen are Triangles with two equal sides, each about eleven inches long, and a base or third side so short [...] With us, as with you, these Triangles are distinguished from others by being called Isosceles.
Rarely - in proportion to the vast numbers of Isosceles births - is a genuine and certifiable Equal-Sided Triangle produced from Isosceles parents

"Is it true? Is it?"

"I know he was born last week, but-"

"Are his sides all equal? All three of them? An actual Regular?"

"That will be for the Board to decide."

"It has been so long since last time I saw a Regular come from our kind!"

"I want to see him! Mom, mom! Can we se-"

"Make way! Make way! Don't you all have duties to return to?"

The small crowd of Isosceles scatters immediately, gone as though it has never been, to make way for the seven members of the Sanitary and Social Board - Squares, for the most part, but it includes a couple of Pentagons and even an Hexagon - stepping before the house.

"Acute-angled rabble," the Hexagon mutters, but truth be told he is pleased to see the Isosceles' excitement over the birth - the possible birth, as it was not verified - of a true Equilateral Triangle from their own. It keeps them hoping, keeps them from despairing. Something to be avoided at all costs, as from desperation comes rebellion, and from rebellion, chaos.

And that, indeed, is to be avoided at all costs.

The occasional emergence of an Equilateral from the ranks of his serf-born ancestors is welcomed, not only by the poor serfs themselves, as a gleam of light and hope shed upon the monotonous squalor of their existence, but also by the Aristocracy at large; for all the higher classes are well aware that these rare phenomena, while they do little or nothing to vulgarize their own privileges, serve as almost useful barrier against revolution from below.

This occurrence - a Regular born from the lowest class - is rare enough, and usually the result of hard work for improvement, generation after generation. Strict control, well-thought-out intermarriages and constant effort to improve the intellect and shape are essential for any line whose desire is to produce, someday, a superior form.

The Sanitary and Social Board sees no evidence of a such work in the Isosceles who opens the door, identifying himself as the father of the infant. He has the humbleness befitting his brethren - a Worker, this one, not a Soldier - but that is all that can be said for him. His base is wretchedly short, the angle all too acute even by his kind's low standards. The wife is - as all females - a straight line. Their words and gestures bespeak the lack of intelligence of the low.

The Board would expect the the offspring of such ill-endowed parents to be itself unfortunate, if not Irregular. And yet, the infant staring up at them - eye wide and wandering, as though trying to take in all that there is to be seen in the barren room - is, quite obviously, not an Irregular.

Careful examination and mensurations, carried on with as much rigour as it is possible while dealing with a child trying to snatch the measuring tape from their hands, confirms as much.

With six votes out of seven, the infant is passed into the class of the Equal-Sided. He also gets to keep the measuring tape, having eaten it as soon as a Square got distracted.

After a strict examination conducted by the Sanitary and Social Board, the infant, if certified as Regular, is with solemn ceremonial admitted into the class of Equilaterals. He is then immediately taken from his proud yet sorrowing parents and adopted by some childless Equilateral, who is bound by oath never to permit the child henceforth to enter his former home or so much as to look upon his relations again, for fear lest the freshly developed organism may, by force of unconscious imitation, fall back again into his hereditary level.

The Equilaterals who obtain custody of the infant are not childless, but may very well be. Their own child - their only son - is an Irregular, none of his sides and angles matching the others. He tends to lean on one side, his gait as irregular as his form as a result. The wretched thing is kept at home, away from scorn and to be more closely observed. It his hoped his irregularity may fix itself over time, before the Inspection, just enough so that he may be allowed to live, even if his place in life will never be less than miserable.

It is a hard life, that of the Irregular, so much that termination is often the kinder option even if dreaded - but if one like that is shown in a positive light, what will become of the Laws of Nature?

He remains out of sight when authorities present his parents with the infant; out of sight is where most of his time is spent. They are both overjoyed to raise a proper child, and promptly pledge to stick to the rules that dictate no contact between him and his family of origins is to take place.

Some jokes are made that they may as well have made a swap, giving poor irregular Liam away in exchange for Bill, this perfect Equilateral that has come from such a low line.

That does not happen. They raise both, although it is clear where their favor lies, and they strive to keep them separated as much as they can, as though afraid their defective offspring may taint this one, and thus both may be lost.

Their attempts are doomed from the start, however. Liam craves companionship.

Bill just craves.