The universe of *Dichronauts*, with two dimensions of space and two of time,
obeys some very strange laws of physics and geometry, which you can read about in this introductory page.
Here, we sketch a possible world within this universe. We won’t call it a “planet”, because that word carries too much
baggage from our own universe.

In everything that follows, we will give the three dimensions of space the coordinates *x*, *y* and *u*, where *x* and *y*
are ordinary, “space-like” dimensions, and *u* is a “time-like” dimension. This does *not* mean that *u*
is the same as our time coordinate, which we will call *t*; it just means that someone else whose state of motion was very different from
our own could have *u* as *their* time coordinate. It also means that when we measure a distance
in our three-dimensional space from the point (0,0,0) to the point (*x*, *y*, *u*), the square of the distance
will be *x*^{2} + *y*^{2} – *u*^{2}, if that expression is positive,
or its opposite, *u*^{2} – *x*^{2} – *y*^{2}, otherwise.
These notions are explained in more detail in the introductory page.

- Gravity, and the Shape of the World
- The Sun, the Habitable Zone, and Absolute Summer
- The Grain of the Land, and the Migration

In our own universe, when the force of gravity shapes a large object, such as a star or a planet, the result is something very close to a sphere. For an idealised object with perfect spherical symmetry, gravity always points towards the centre of the sphere, and the whole surface is at the same gravitational potential. This is a stable configuration, at least so long as the material in the interior of the object is strong enough to resist the pressure of all the weight overhead.

In the *Dichronauts* universe, the shape that is analogous to the surface of a sphere is a hyperboloid. There are two distinct kinds of hyperboloid:
one-sheeted and two-sheeted (red and green, in the **image on the right**). The first kind is a bit like an infinite hour-glass; the second kind is like a pair of infinite bowls facing in opposite directions. In the picture, of course, we can only show part of each infinite surface.

What we need is a solid, three-dimensional object whose surface — or surfaces — are
one or more hyperboloids. Such an object will be perfectly symmetrical about its centre, in the *Dichronauts* geometry, in the same way that a sphere
is perfectly symmetrical in Euclidean space: if we rotate the object in any way at all that keeps its centre fixed, it won’t look
any different. (If that sounds confusing, read the section in the introductory page on geometry and rotations.)

Any such object will be infinite, with an infinite volume and an infinite total mass. We
can imagine cutting off the hyperboloids at some point, truncating the object to obtain something finite; that would spoil its perfect
symmetry, but physical objects are rarely exactly symmetrical. However, the perfectly symmetrical,
infinite versions have the advantage of being simpler to describe mathematically — and in fact, so long as no locally measurable physical
quantity (such as the strength of gravity, or the pressure in the interior of the object) is infinite, we can even entertain the possibility of such things
existing, in the hypothetical *Dichronauts* universe.

The mathematical law obeyed by the force of gravity in our own universe is, famously, an inverse-square law: the force between two point
masses is proportional to both masses, and inversely proportional to the square of the distance between them. In the *Dichronauts* universe, the
analogous law is identical, except that “the square of the distance” now means *x*^{2} + *y*^{2} – *u*^{2}, if that expression is positive,
or its opposite, *u*^{2} – *x*^{2} – *y*^{2}, otherwise.

Of course, there will be a cone on which *x*^{2} + *y*^{2} – *u*^{2} = 0,
so the gravitational force from a 0-dimensional point mass will be infinite on that cone. This is somewhat troubling, and more so than
the fact that the gravitational force from a point mass in our own universe grows infinite as you approach the point mass itself.

In terms of the microscopic constituents of matter in the *Dichronauts* universe, this suggests that we will need to avoid any
literally point-like particles, and replace them with objects where the mass (and electrical charge, if any) is spread out over a finite
region. But we don’t need to delve into the details of particle physics to side-step this problem; both in our own universe and the
*Dichronauts* universe, it’s easy to express the law of gravity in terms of the overall *density* of matter, rather than in terms of the
mass of individual particles.

In the *Dichronauts* universe, the appropriate law is this:

If we add up the second rate of change of the gravitational potential with respect to each space-like coordinate, and subtract the same quantity with respect to the time-like coordinate, the result is equal to 4 πGtimes the density of matter, whereGis a constant that describes the strength of gravity.

As you can probably guess, the only difference in our universe is that we add together the second rate of change of the potential with
respect to all three coordinates, *x*, *y* and *z*; nothing is subtracted. This way
of formulating Newtonian gravity can be interpreted as a geometrical statement about the “lines of force”
that can be used to describe the gravitational field of an object. It amounts to saying that these lines of force only start and
end in matter (never in vacuum), and that the density of the endpoints of the lines is proportional to the density of matter.

So, we should look for a gravitational potential that obeys this law, for a perfectly symmetrical arrangement of matter in the
*Dichronauts* universe. For the sake of simplicity, we will assume that the density of matter is constant throughout the interior of
the world.

There are three fairly simple ways in which we could obtain a perfectly symmetrical arrangement of matter. We could take either the one-sheeted or the two-sheeted hyperboloid, and completely fill the space bounded by the surface. But in either case we would be piling up an infinite amount of matter in some directions.

The third way would be to take both kinds of hyperboloid, and fill only the space between them, as in
**the image on the left**. This is still an infinite object (although the picture just shows a finite piece),
but if we choose a small region anywhere on the surface of one of the hyperboloids and look at the volume of material below it, right down to the
centre of the world, the
volume will be finite, and will depend only on the area of the region and the radius of the hyperboloid.

The gravitational potential that corresponds to this kind of distribution of matter
is plotted in the **image on the right**.
The plot is for a slice in the *x*-*u* plane, but if we rotated the slice around the *u*-axis by any angle,
the potential would look exactly the same.

This potential corresponds to a gravitational force that always points directly towards the centre of the world.
That might seem surprising, given the way it slopes *down* from the centre as we move along the *u*-axis,
but because kinetic energy is negative for objects moving in time-like directions, anything dropped in this region will “roll uphill”
and move in the direction of increasing potential. So, gravity on both the one-sheeted and two-sheeted hyperboloids will
be a constant downwards force across the entire surface.

(The mathematical formula for the potential is described in detail in the supplementary page for this topic.)

What kinds of worlds would form in a universe with two space-like and two time-like dimensions? That would depend on
the details of the cosmology, and the constituents of matter. Matter in the *Dichronauts* universe is harder to deal with mathematically
than our own kind; if we assume that it is composed of point-like particles carrying some kind of charge, any force that goes to infinity as the distance
between two particles gets smaller
will be singular on the entire cone for each particle (the surface on which the distance from the particle is zero).
In principle, we could spread out the charge
on each particle over a three-dimensional region, rather than having it concentrated in a point, and that would remove the singularity on the cone. That’s essentially what we did in the previous section when dealing with the gravitational field.
But to predict the properties of this universe’s equivalent of atoms and molecules, we would need to apply quantum mechanics to
individual, extended objects in a space with a very different geometry than our own, which would not be a trivial exercise.

So, rather than attempting to deduce the detailed particle physics, chemistry, and cosmological history of this universe, we will just sketch one possible world, and describe some of the properties of matter that would be needed to make it work.

We will suppose we have a large, solid, hyperboloidal world. As described in the previous section, even if this world is infinite, it will have a finite gravitational field everywhere, pointing towards the centre.

If the solid ground is blanketed entirely in some kind of “atmosphere”, it will be a liquid, not a gas, for reasons discussed earlier. In the world of the novel, such a “liquid atmosphere” is present, but it is sufficiently rarefied, and transparent at visible wavelengths, that to the characters it is much like our own air. A “rarefied liquid” sounds odd, but the long-range forces that stretch out along the cone of each particle would make it possible for a system with relatively low density to have thermodynamic properties more like those of a liquid than a gas.

What happens at the interface between such a liquid and the vacuum of space? While remaining agnostic about the fine details of intermolecular forces, in general there are two possibilities: the liquid could maintain its integrity, the way some solids exposed to vacuum do, or it could undergo a violent phase change into a “cone plasma”, in which particles collide at random and build up huge velocities.

For the world’s atmosphere, we would need a liquid that behaved in the first way, sedately meeting the vacuum, sealed by its own cohesive strength.
But the second possibility would offer a means by which a body of liquid could behave as a source of light and heat. In other words, a “star”.
In our universe, stars need to be of at least a certain mass, sufficient for the pressure at their core to ignite fusion reactions.
In the *Dichronauts* universe, they would merely need to be composed of the right kind of liquid.

The hyperboloidal world of the novel (portrayed in the **image on the left**) is accompanied by a much smaller star, which moves around it in a tight, circular orbit.
This is certainly not the only configuration possible, but the striking differences from our own world, and some exotic
geometric effects that arise in this situation, make for a more interesting story than that of yet another small planet
moving around a large, distant star.

As previously described, any light source in the *Dichronauts* universe has a “dark cone”
into which no light is emitted. So the region of the world illuminated by the sun is bounded in two ways. There is, of course, the usual kind
of border between day and night. But even on the day side, as you move
north and south of the circle over which the sun orbits, you will eventually enter the sun’s dark cone.

You might think that living north or south of the circles where the dark cone first intersects the world
would merely result in a curious phenomenon where the light vanished around noon, with this odd midday blackout growing longer
the farther north or south you went. But this is where the geometry starts to bite. Near the sun’s dark cone, the distance
from the surface of the world to the surface of the sun comes ever closer to zero, and the intensity of solar radiation striking the ground
becomes extremely large. Standing in a place where this happens, even briefly, twice a day, would be fatal. So these circles mark
the boundary of what is
known in the novel as **absolute summer**: those regions where the distance to the sun is sometimes zero, and there is no possibility
of surviving.

In fact, in contrast to our own world’s correlation of high latitude with cold, the coolest place on this world will be
the circle directly below the sun’s orbit, which is called **the midwinter circle**. Every other point is actually closer to the
sun at noon, as measured by the geometry of this universe. Stretching north and south of the midwinter circle is **the habitable zone**,
where the temperature is hospitable to life. This ends well short of absolute summer, once the temperature becomes high enough for
“water” (not water as we know it, but whatever ubiquitous solvent serves a similar purpose for life on this world) to
start dissolving into the liquid atmosphere so rapidly as to leave the ground parched.

In the image, the boundaries of the habitable zone are marked by blue bands of shallow swampland, where the drop in temperature first allows the moisture locked up in the warmer air to the north and south to fall as rain.

Suppose we arrange a number of geometric shapes in a grid, and draw an arrow on all of them,
pointing in the same direction. We then allow the shapes to rotate at random, limited only by the
fact that they might bump into their neighbours if they turn too far. If we don’t pack the shapes together too tightly, then in Euclidean space, the arrows can end up pointing in random directions, as in **the top image on the right**.

But if the same grid of shapes lies in a plane in the *Dichronauts* space with one space-like and one time-like dimension, the
way objects stretch out as they rotate will cause them to bump into their neighbours as soon as they’ve turned
through a very small angle, and
the end result will be more like **the bottom image on the right**.

This example is a bit artificial, but it demonstrates a general principle: on the surface of the one-sheeted
hyperboloid — where one of the horizontal directions is space-like and the other is time-like — the ground will exhibit
a strongly directional *grain*, in a way that the surface
of the Earth does not.

If you were taken to an unfamiliar location on the surface of the Earth on a cloudy night, and equipped with
powerful artificial light that let you examine the ground as closely as you liked, you would have no way at all
of determining which directions were north, south, east and west. There simply is no reason why any geological feature
should encode that information. But on the one-sheeted hyperboloid of the *Dichronauts* world, not only is there a
fundamental geometrical difference between north-south and east-west (indeed, you can’t even see by light
in any direction within forty-five degrees of north or south), but even grains of sand will not be free to lie at angles
that differ too much from their neighbours, because if they tried to adopt their own, idiosyncratic orientation they would
be blocked by all the grains around them.

We have described the hyperboloidal *Dichronauts* world as being perfectly symmetrical — unchanged by any rotation that
leaves its centre fixed — but of course that is only true of an idealised, smooth, homogenous version of the world,
good enough for computing its gravitational field, but not for understanding what it would be like to walk around
on the surface. Just as the
Earth has a rugged topography of mountains and valleys, so will the *Dichronauts* world. But while an explorer almost
anywhere on Earth — unless they are standing on the narrowest mountain ledge, or squeezing through the tightest
crevice — is free to turn their body three-hundred-and-sixty degrees, the body of any creature on the
*Dichronauts* world will be forced to conform to the geology. Alone on a vast, smooth plane, the effects of rotation
would only impose the strict geometrical law that no space-like direction can rotate into a time-like one, or *vice versa*, but as soon as there
are obstacles protruding from the ground, the need to avoid colliding with them begins to intrude, and most kinds of
landscape would make it difficult or impossible to go against the grain.

So, instead of treating the surface of the *Dichronauts* world as a featureless hyperboloid,
we should think of it as endowed with a grid of **geological coordinates**, with longitude wrapping around the
hyperboloid, and latitude running up and down from some equatorial circle. But while the coordinates we use on Earth
are locked to its rotational poles, the *Dichronauts* world does not get its days by rotating on an axis, and
it does not have a unique geometric axis, any more than a sphere does.
(When we draw it in Euclidean space, it has a unique axis of symmetry, but that’s just an artifact of
the geometry we are using to visualise it.) This leaves us free to choose the geological coordinates in a way that
reflects the grain of the land: geological north, south, east and west are the (average) directions in which you
could orient
the sides of a square box and place it on the ground without it bumping into things. On Earth, that
definition would be useless: you could turn the box to face any way you liked. But on the *Dichronauts* world,
you turn a square too far at your peril.

There is, of course, another natural system of coordinates: one that is determined by the
sun. The midwinter circle — the circle that lies on the surface of the world directly
below the orbit of the sun — would make an obvious choice for the equator in a system of *solar*
coordinates.

Is there any reason why these two systems of latitude and longitude should agree? It is certainly possible to imagine that the world and the sun formed at the same time, and were governed by some shared conditions that brought them into alignment. But it is also possible to imagine them being independent.

In the story of the novel, there is a further twist: although geological and solar directions might have been the same in the distant past, the relationship between them has been changing for eons. It is a matter of contention whether the world is slowly rotating relative to the fixed orbit of the sun, or whether the world is fixed and the orbit of the sun is slowly tilting, but in either case the effect is the same: the habitable zone is constantly moving across the land, forcing people to move with it in a slow, but unceasing, migration.

For the sake of convenience, we will describe what follows as if it is *the world* that is rotating.
This rotation involves one space-like and one time-like dimension, so it
is not a circular, periodic motion like the sun going around in its orbit.

In the **image on the left** we see two views of the surface of the hyperboloid (to which the red and blue
**geological coordinate grid** is fixed) rotating out of alignment with the **solar coordinate grid** (in grey). The habitable zone,
which is fixed to the solar coordinates, is in green.

The leftmost view faces one of the **nodes** — the points on the surface left unchanged by the rotation —
while the second view is at ninety degrees to the first.

Close to the nodes, the effect is essentially a two-dimensional rotation of the surface around the node. If someone was standing here, and they kept the orientation of their body fixed relative to the landscape around them, they would eventually rotate so far, relative to the habitable zone, that their body would protrude out of the zone and into absolute summer. Or, to describe it from their point of view, the habitable zone would shrink into something narrower, in one direction, than their body. If the ground was perfectly smooth and flat, they could save themself by turning in a way that kept their orientation fixed relative to the habitable zone. But in any realistic landscape, that would eventually become impossible.

Ninety degrees away from the node, this problem vanishes. Although the land keeps moving north with respect to the sun, making it necessary to migrate southwards in order to stay in the habitable zone, the geological and solar definitions of east-west and north-south remain in agreement.

Now, at some point between these two locations, the disagreement between solar and geological directions is going to start to cause trouble. The question is, does the strip of land within the habitable zone that is trouble-free keep shrinking indefinitely as the world keeps rotating, or does it converge on some fixed width that will remain essentially unchanged over the eons?

The answer can be seen in **the image on the right**, and it is good news for inhabitants of this world.
This plot shows a parameter which measures how far the geological and solar directions are rotated relative to
each other, across a range of longitudes measured from one of the nodes. The different grey curves are for successive
values of this parameter at the node, while the red curve shows what happens as the rotation at the node
goes to infinity. This limiting curve shows that there is always a significant range of longitudes where the relative
rotation is not too large, and within this region
people will always be able to orient their bodies to conform with the grain of the land,
without any risk that they will protrude out of the habitable zone.

(The mathematics behind these curves is described in detail in the supplementary page for this topic.)