: Inspired by Edwin Abott’s Flatland

Collected below are the notes for an unfinished travel memoir penned by the explorer and cosmic cartographer Helkanoum. I have here attempted to organize them into a coherent narrative, but so much of the material has been lost that we can only guess at the true scope and ambition of Helkanoum’s intended work.

Preface

1.1.1 I am the planar one you have heard so much about.

1.1.2 … an account of my life, published in your tongue with the title “The Flat One Who Crawled Up”. It contains many factual inaccuracies with respect to the physics of my world, and the societies contained within it. I will address these in detail in the following pages …

1.1.3 I hope that this text will go a little way to bridging the gap between our spaces. It is necessary now to understand and embrace one another, as the many-sided infinity we live in is no longer folded to keep our races apart.

Introduction

2.1 The basics must be familiar to you. My universe spans an infinite area, but holds no volume. My species can recognize breadth and length by sense experience, but until my voyage, the possibility of a third dimension was treated only in certain speculative branches of physics. As of the writing of this book, I am the only one to have ever seen “up” and “down” first hand. Misinformers may have told you that my world is a blank and featureless sheet, and that we are race of straight-edged polygons – and futhermore, that we are not merely simple-bodied, but also simple-minded. This is a falsehood. My world is as richly stocked with irregularities, curvatures, spirals, color, texture, light, life and intelligence as yours.

2.2 The affairs of two dimensions keep my mind so occupied as it is, that I have no desire to master the goings-on of a third. And so I have returned to my own realm and to a peaceful retirement.
2.3 I was once the prime admiral and cosmographer of all the flat worlds, but I have aged, and perhaps I’ve gorged myself on infinities for too long. I’ve lost the stomach for it. The time has come for me to shut away my instruments, and make a record of my findings before I die. I write to you of the worlds I have seen, and what I understood of them.

2.4 I speak for the point, line and sheet beings of my world, and for planar life-forms generally, wherever they reside. To you, I say: we are serene. We glory in our arts and sciences, which are without equal in all the hyperspaces.

2.5 In recent times, certain high-dimensional beings have made a mockery of our religion. They think they awe us with demonstrations of godhood, acts of teleportation, transmutation, omniscience and omnipresence. We are not made devotees so easily. When these ill-wrought miracles are greeted with scorn, they brand us dim-witted. They say we are disrespectful to Heaven. We are not stupid, we merely see these tricks for the banalities of motion and extension they represent. For a three dimensional being, the ability to vanish and then reappear in another place in our world, or at several places at once, is a simple consequence of our inability to detect your movement and true shape in the vertical realm. I too could poke through a lineword at multiple points and terrorize its inhabitants with illusions of a similar description. But I choose to make better use of my time.

2.6 Though we do not have a sense perception of it, we comprehend the third dimension perfectly. Our mathematical intuition is the envy of all the hyperspaces. The power of abstract thought is surely diminished in minds that must contend with stimuli from three, four or more dimensions. Flat life-forms make, on average, better geometricians than those races who know – by direct sense experience – how to pull 12 arrows from a point, and have each one be at right angles to every other one.

On the Flat Universe known as Yaroshtebar

3.1 I am committed to recording my cosmographical findings for all sentient life, including you – my dear hyperdimensional reader. There are many flat worlds – mine is not the only one. I am a cosmographer, and I have dedicated my life to mapping them and understanding their physics. And so I will speak to you of the birth, evolution and death of one such cosmos I visited in my youth, which I have named Yaroshtebar. For those of you who are etymologically inclined: Yaroshtebar is a game of chance we play at intervals in the life of a community. In it, a body of citizens assembles and two die are cast, one black and one red. If the black dice rolls higher, we retire home to an evening of quiet meditation. If the red roll is higher, a night of debauchery must ensue. And so we let, for the course of an evening, the goddess of chance dictate our social affairs.

3.2 When I set to the task of building a map of a cosmos, I begin with what is absolutely elementary. My mind sets the clock and my body serves as the measuring stick. In Yaroshtebar my whole body is extended over a single point. Space appears to me as a grid of discrete points. Displacement is the smallest number of points between two points. And so I am 12 points away from A, as the arrow flies. 13 points to B. 4 points between A and B. Now we are three points in a triangle. I move. A is now 13 points away, B, 14. The separation between A and B remains undisturbed. Thus a new state of affairs is born that was not in place before. Time is the thing that allows change. In the universe of Yaroshtebar, time is organized into discrete steps, during which each object is permitted a single motion.

3.3 Timeless universes are in perfect stasis. The stream of consciousness is halted immediately when a cosmographer stumbles inside one. I avoid them at all costs. Some maintain that time exists, even in a frozen universe where no objects are in motion. I abandon this weighty issue to the philosophers.

3.4  … Yet there are even more dimensionally-impoverished beings: those who dwell in point universes, in which objects have neither extension, position or separation, but all matter is coincident in space at a single location. But, I regard them to be the richest and purest creatures in existence. As space is all balled up in a single coordinate, so too are their spirits overlapped in a perfect and blissful communion, free of the divisions introduced by side, area and volume.

3.5 What is matter in Yaroshtebar? A point in space can be empty, or occupied by matter. Law #1 Only one unit of matter can reside in a point, and no more. A unit of matter I shall call a particle. There are two types of particles, red and black.

3.6 Law #2 When a red particle impinges on a point occupied by a black particle, or vice versa, they are both annihilated. A third law comes from close monitoring of the total mass contained in this world. My measurements imply that (Law # 3) this universe cannot support more than 200 particles – 100 red, 100 black. When the amount of matter is depleted below this limit, it is replenished by a generative process that creates matter from vacuum. What is this generative process?

3,7. Law #4: At each time step, at a given empty point in space, a particle may arise from vacuum with a certain probability P(G). The equation for P(G) is given below:

Where Ri is the displacement between a point and particle i (out of n particles). No set of 100 Ri values in this finite space will allow this probability to exceed one.  All red particles provide some contribution to the probability that another red particle arises. Black particles make no contribution to the probability that a red particle will arise. The same principle applies to black particles. Red matter begets red matter and black begets black.

The probability that a red particle will arise at a given point in space increases if there are many red particles that are positioned nearby (Ri is small). If they are distantly placed, or if – at a given moment – there are no red particles in the universe, then Pr(Generate Particle) reduces to the base probability 0.00001.

1.1.4 If these stipulations seem arbitrary and strange to you – I urge you to consider how arbitrary and inane the constants and laws of your universe are. Why is the matter-limit rule of this world any more unusual than the Law of Conservation of Energy in yours?

Now let us consider motion. A type of gravity can be observed in this world that determines how particles move. Law #5 At each time step, a given particle can move north with a probability Pn (or south with probability (1-Pn)). In the absence of any other mass in the universe, the probability of moving north or south is the same: 0.5 (see equation below). These probabilities are biased away from 0.5 in the presence of matter, depending on the number of particles in the universe, and their position relative to the particle being investigated.

Where Ryi is the vertical distance along the y axis to particle i (out of n partcles)  of the same type , and ryj is the vertical  distance to particle j of the opposite type. (along the y axis). A consequence of this law is that a particle is more strongly attracted to matter of its own type, than it is to the other type – note the factor of 0.75 multiplied to the sum of distances to dissimilar particles. Red attracts red more strongly than it attracts black. The same rule applies every time when considering East-West motion as well.

Rr(East) is (1-Pr(West). Rxi and Ryi are the horizontal distances to particles i and j of the same and the opposite matter-type.

The universe of Yaroshtebar was empty in the beginning. Then red and black particles arose. The particles aggregated into a clumpy grid of stars. With time, The stars coalesced or annihilated each other until only two large stars remained, one red and one black. Eventually the red and black star collided, and a stable and eternal balance of constant annihilation and regeneration arose due to the contest between laws #4 and #2.

(I depict this progress in the two films below. Time is sped up towards the end of the animation (GIF/movie available)).  –