## Naming of 4D cardinal directions

Ideas about how a world with more than three spatial dimensions would work - what laws of physics would be needed, how things would be built, how people would do things and so on.

### Naming of 4D cardinal directions

In 3D, we have the following relative directions, i.e., they are relative to the coordinate system of the observer: forwards/backwards, left/right, up/down. We also have the following absolute directions, i.e., they are relative to a fixed reference frame common to all observers (on a 4D planet, say): north/south, east/west, up/down.

In 4D we need an extra pair of directional names for the extra dimension in each category. What are the best terms to use? There's ana/kata, but it's unclear whether it refers to relative directions (relative to the observer/speaker/etc.), or an absolute direction (relative to a hypothetical 4D planet, say). There's also marp/garp, which sounds a bit silly and I'm not a fan of it.

What other words can we use/coin for this purpose?

(And yes, I'm aware that the absolute category may have two divergent definitions, depending on which model of 4D planet you subscribe to. But I'm looking for good words to use to designate directions in either system of reference directions, not so much in the directions themselves.)

(Also, I'm aware that up/down are not distinguished in 3D, mainly because we generally stand upright on the surface of the planet, and rarely orient ourselves any other way, so relative up/down almost always corresponds to absolute up/down. For the purposes of this discussion, I'm assuming the same for 4D.)
quickfur
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### Re: Naming of 4D cardinal directions

P.S. I'm also looking for words that are amenable to adjectival derivatives, e.g., north -> northern, northerner, etc.,; east -> eastern, easterner; ana -> ?, ?; kata -> ?, ?
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### Re: Naming of 4D cardinal directions

Many years ago on the 4D Wellston Forum I remember Jonathan Bowers AKA Polyhedron Dude had mentioned Wint and Zant.

From http://hi.gher.space/classic/glossary.htm

wint adverb [Jonathan Bowers] - One of the two extra turning directions of a tetronian object. In tetraspace, represented by the vector <0,0,0,1>. In realmspace there are two turning directions, left and right, but in tetraspace, there are two more. The four tetronian turning directions are located in a particular order depending on which direction you look at them. If you look at an object containing these directions and you look from the back of the object towards the front, then when you traverse them in a clockwise direction starting from the left, the order is left, wint, right, zant. Thus, wint is 90 degrees clockwise from left and 90 degrees counter-clockwise from right. Zant is the opposite: 90 degrees counter-clockwise from left and 90 degrees clockwise from right. See the chart under direction.

marp noun [Jonathan Bowers] - The cardinal direction in the direction of polar rotation (rotation along the polar equator). The opposite direction is garp. Distance towards marp or garp is measured in laptitude.

delta adverb - One of the two directions pointing out of realmspace into tetraspace; movement in the negative w direction. In tetraspace, represented by the vector <0,0,0,-1>. This term refers to the direction analogous to down; the 'd' in 'delta' corresponds to the 'd' in 'down'. In some literature, this direction is referred to as ana. This term is the same direction as zant, but is used in a different context: delta is movement from realmspace to tetraspace, but zant is a turning direction.The opposite direction is upsilon. See the chart under direction.

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### Re: Naming of 4D cardinal directions

I discuss this topic in my forthcoming book. In short,

I call the two "poles" the Red Circle and the Blue Circle. The Red gets more sunlight so it is a hot red desert like Moab Utah or Mars, the Blue is cold with oceans like Patagonia. "Red" is the direction of minimal distance to the Red Circle while Blue is the opposite, which is also the minimal distance to the Blue Circle. The other four directions are then constrained to be perpendicular to Red-Blue. East is parallel to the Red Circle and in the direction of Red rotation, West is the opposite. The remaining two directions are parallel to the Blue Circle. I whimsically call them Tar and Zana after the Los Angeles suburb of Tarzana. Tar is in the direction of the Blue rotation, Zana the opposite.

The Red plane rotates faster, so sunrise tends to be in this direction. That's why the Red direction of rotation is called East. On average, the sun rises somewhat to the Tar of East.

Mechanical clocks have two hour hands, one red and the other blue. The periods of the two hands correspond to the periods of the two rotational planes. These hands are mechanically connected in a simple way. The length of each hand depends on the latitude. This forms an analog computer that approximates the height of the sun above or below the horizon. This is rather more complicated than on our Earth. It's particularly useful for showing/predicting sunrise and sunset.

In a planet with a Clifford rotation a 3D Earth style clock will work just fine. There are no distinct planes of rotation so pick two arbitrary perpendicular rotational planes to be Red and Blue. I suppose one would intersect some landmark like the Greenwich observatory.
Last edited by PatrickPowers on Sat Jan 30, 2021 6:38 am, edited 2 times in total.
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### Re: Naming of 4D cardinal directions

PatrickPowers wrote:In a planet with a Clifford rotation a 3D Earth style clock will work just fine. There are no distinct planes of rotation so pick two arbitrary perpendicular rotational planes to be Red and Blue.

It's not quite arbitrary, even when we require the planes to pass through the centre of the planet.

Pick any unit vector, representing a location on the planet (with 3 degrees of freedom), and the direction of rotation gives a perpendicular vector. The product of these two vectors gives a plane of rotation. These orthonormal vectors can rotate together in the plane, while their product remains unchanged; this removes 1 degree of freedom. (All points on the circle are equivalent.) Then there's a unique plane orthogonal to the first plane, and we can take this pair to be Red and Blue. This construction has 2 degrees of freedom.

An arbitrary plane, and thus an arbitrary pair of orthogonal planes, has 4 degrees of freedom.

Of course, if it's not an isoclinic (Clifford) rotation, there are no degrees of freedom; there's a unique pair of planes of rotation. An arbitrary point on the planet (not on one of these planes) rotates not along a circle, but along a kind of helix; the product of the two vectors doesn't give a plane of rotation.
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mr_e_man
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### Re: Naming of 4D cardinal directions

That's right, if the planet is in an isoclinic rotation, every fixed point on the surface rotates along a fixed circle that defines a unique plane of rotation in which it rotates. The planes defined by the other points trace out circles that swirl around this fixed circle (this is the Hopf fibration of the 3-sphere), with a unique orthogonal circle that defines an orthogonal plane. You cannot simply pick any pair of orthogonal planes, because all the points that lie in any one of the planes must belong to the same orbit under the isoclinic rotation, otherwise it's not a plane of rotation. There are an infinite number of legal pairs, but these are a subset of the set of all pairs of planes.
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### Re: Naming of 4D cardinal directions

What do you mean, "depending on which model of 4D planet you subscribe to?"
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### Re: Naming of 4D cardinal directions

개구리 wrote:What do you mean, "depending on which model of 4D planet you subscribe to?"

There are (at least) two possible models of a 4D planet:

(1) The "traditional" model, that draws from direct dimensional analogy from 3D planets modelled as a 3D sphere, in which there's a north pole and south pole, and an equator that lies midway between the poles. According to this model, the equator would span a 2D spherical region. So "north" and "south" would refer to unique directions towards either pole, and the remaining directions would lie parallel to the equatorial sphere.

(2) The "toroidal" model: this is based on the Hopf fibration of the 3-sphere, under the assumption that a 4D planet with non-zero angular momentum would likely equillibrize into a Clifford isoclinic rotation. Such a planet would not have any fixed poles that could be designated as north or south poles; instead, it would be divided into toroidal zones based on the way the star it orbits appears in its sky. The great circle closest to its orbital plane would see the sun rise more-or-less directly overhead at daytime, and the toroidal region around this great circle would have equatorial climate. The further one moves away from the equatorial great circle, the lower the maximum height of the sun would be; eventually, one converges onto the orthogonal great circle, where the sun basically remains close to the horizon in a perpetual dusk. This great circle would be a "polar circle", a different analogy from 3D: a 3D planet's poles can be considered to be the points of maximum distance from the equator, which in 3D is two points we call the north/south poles. In 4D, the points of maximum distance from the equator form an orthogonal great circle, so we may consider this circle as the "polar circle" (arctic circle? With a different meaning, though). The region between the equatorial circle and the polar circle would be a region of temperate climate.

Under (2), the system of directions would be drastically different from (1); "north" and "south" would cease to refer to fixed reference points, but would refer to geodesics towards either the equatorial circle or the arctic circle. North/south would no longer cross the equator; instead, north would diverge from the equator and converge on the polar circle, whereas south would converge on the equator. East/west then would correspond with the direction of sunrise/sunset over the equatorial circle, and two other directions would be needed to designate the remaining dimension of motion perpendicular to these.
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### Re: Naming of 4D cardinal directions

In four dimensions, the names are up/down (gravity), forward/back (motion), but I am afraid you can't divvy up the remaining two. All you can do is give parity to a circular motion.

In regards to the planet, isoclinal or clifford rotation is most likely when energy is exchanged with rotation.

The model with seasons works if you suppose that the sun is not following an isocline, but is on a circle that is of opposite parity to the rotation.

What happens then is that the sun is over a point that is some fixed distant from the common parallel. As the year passes, all parts of this torus will have the sun directly overhead, so you have things like the tropic of Leo and the tropic of Taurus, etc.

The east-west direction is defined by the stars, that is, the zenith-star always rises in the east-most point and sets in the west-most point. The position of the sun is what sets the day time.

Note that the sun in the sky moves in a circle in the sky, and this circle maps onto a circle on the lattitude sphere. This gives rise both to the seasons and the climata. The lattitude sphere at any given point can be derived from taking a sphere perpendicular to the E/W axis, the axis running from the zenith to the observer. The sphere is identical for all parts of the world, but the zenith point is varyingly different. In other words, each point on the lattitude is a great arrow (circle + direction) on the glome.

Since the year-path of the sun is a line of lattitude on this sphere (ie 47 degrees from the S pole), we get the south pole = 0 degrees, the tropics at 23.5 degrees (where the sun comes overhead), the artics at 67.5 (where the sun goes to the horizon), and the north circle (where the sun rises always to 23.5 degrees at noon,

As the sun follows its track, different great circles in the tropics are brought under the sun, and the line running from the Npole to Spole on the lattitude-sphere represent the places where the sun cumulates. That is, this line on the lattitude circle represents the highest rising of the sun, it moves around the planet as the sun does. The seasons lag due to momenta, but everywhere there is somewhere in spring, somewhare in autumn, etc, roughly by imagining that the longitude-lines on the lattitude-sphere are not times of day, but months of the notional year. You have season-zones as we have timezones.

In the ordinary 3d earth, the climata runs from the equator to the poles, and the two axies are moving at 6 months apart on the year-circle. In 4d, you have a full disk, where the equator is the co=parallel nearest the tropics, is S, (or the 'equatorial climates), and the one further distant is the N.

The seasons are not caused by the lengthening of the day, but how far the sun rises. Imagine a sphere, with the line of 43 S picked out on it. This is twice 23.5 degrees from the south pole, and represents the tropics. If you place this sphere on the ground, it will essentially represent the lattitude sphere at the current point. The circle of 43 degrees S, marked out in the months, represent points where the sun will cumulate. In essence, a line drawn from where this sphere rests through the ground, through the point on the sphere, will strike the sky on the E/W divide, where the sun will rise to on that day.

A sun-dial cut into this sphere, will exactly show the time if the N/S axis is correct, and set perpendicular to the E/W axis.
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### Re: Naming of 4D cardinal directions

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### Re: Naming of 4D cardinal directions

The seasons on a planet with an isoclinal rotation or nearly isoclinal rotation can be quite exotic and unlike anything we know here on Earth. What complicates things is that the ecliptic -- the plane in which the planet orbits its Sun -- may have any relation to the rotation of the planet. It's that way even in our Solar System, where the ecliptic is almost perpendicular to the rotational plane of Uranus. So there are all sorts of possibilites.

An extreme is when the ecliptic coincides with a rotational plane of the planet. Then that plane gets maximum sunlight every day. On the other hand, the rotation plane perpendicular to that sees the sun on the horizon at all times. If there were even modest hills in this area the Sun would never ever rise. It would be twilight all the time, a true Twilight Zone. Such a planet has no seasons, only climate. There is a bright hot ring, a cold dark ring, and transition in between depending on how far one is from that special rotational plane.

The probability of a permanent twilight zone seems quite low. But no matter what the ecliptic is, we can say that every day this occurs somewhere on an isoclinally rotating planet. There is always a place somewhere where the sun remains all the day on the horizon with half of it showing. At sea this would be quite apparent.

Usually the ecliptic will not lie close to a plane of rotation. Seasonal changes result, but they are of an exotic form. In tropical regions these seasons are extreme, rather like the seasons we have at the poles of Earth. Each season has a zone that moves about on the surface of the planet. It is always spring, fall, winter, and summer in various places. The further one gets from the tropics the less effect have these exotic seasons. Instead one sees temperate latitude seasons similar to ours.

On all 4D planets both types of seasons go through their cycle twice in an orbital year. If you want an Earthlike planet then the orbital year would be 730.5 days, equal to two seasonal years. A month remains about 30 days. Twelve months in a seasonal year, 24 months in an orbital year.
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### Re: Naming of 4D cardinal directions

I did all these calculations a year ago, this discussion has inspired me to revisit the subject.

It comes down to this. Here on 3D Earth the zone of minimal climactic variation and the plane of the rotation are the same. In 4D this is not so: there is no connection between the two. So which to choose?

On a planet with planes of rotation with different periods then basing things on planes of rotation is the choice. On planets isoclinal rotations this would not arise naturally. There are no special planess of rotation. The exotic "phase seasons" would dominate life, so the clear choice would be to chose the the cardinal directions to correspond to those. This means basing the cardinal directions on the ecliptic. I haven't worked it out yet but it will go something like this.

Consider the concept of a great torus. Tori composed of great circles on the surface of a sphere are of different sizes. A great torus is such of torus but of maximum size. Such is known as a "square flat torus" because it can be flattened into a square without any distortion. Non-great tori can be flattened too but into rectangles. In either scheme the Equator is a great torus, a square flat torus. The question is which to chose. On an isoclinal planet we choose as the Equator the great torus such that the Sun is above it half the year and below it the other half. This then forces the "poles" to be the two circles each at maximum minimal distance from the Equator. (I think these circles are unique, but need to check.) Then the mapping procedes the same way it does on a heteroperiodic planet. The latitude is the distance from the Equator. The longitudes are measured relative to those two circles. From any point of the planet find the closest point on each circle. These are your two longitudes.

Now comes the interesting part. These two longitures are NOT natural cardinal directions. What is of interest as far as the phase seasons go is the difference between the longitudes. You could call it the direction of isophase. Traveling in this direction does not change the phase season in which one finds oneself. The third direction is perpendicular to the other two.

Now I suspect that the result will be an approximation. It will be close enough only if the distance between the 4D Tropics of Cancer and Capricorn is not too large. If it is too large then to get an exact solution you would have to have cardinal directions that are not geodesics. I intend to work this out but it is going to take a while. If all these suppositions are correct then to get an exact solution I believe it will be necessary to have a system without fixed orthogonal coordinates. I'm going to go out on a limb and speculate that there will be one great circle coordinate while the other two cardinal directions will be shaped like the edge of a Pringle's potato chip with circumference equal to that of a great circle. We'll see. If it all works out then we have a planet on which coordinates that are only locally orthogonal are there at the very beginning of astronomy instead of coming in two millenia later. This means they would have a very different sort of geometry. They'd still have Euclidian geometry for local things, building houses and measuring plots of land and so forth, it would definitely be first, but geometry would take a different path much earlier than it did here on Earth. Just what would that primitive HyperEarth geometry be? It seems like it would be a conformal mapping having to do with the Hopf fibration. If anyone has a hint I'm all ears...uh...eyes.
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### Re: Naming of 4D cardinal directions

So what sort of cardinal directions would arise naturally on a planet with an isoclinic rotation? There are no fixed points in the sky, and no fixed points in the magnetic field, so there is no such thing as north or south. Instead almost every heavenly body appears to move in a helix that revolves around East, the direction in which the observer is moving due to the rotation of the planet. The exceptions are heavenly bodies that rise directly to the East. The radius of the helix is too small to see so they appear to be rising in an arc that passes directly overhead.

So, how easy would it be to get a precise direction of East with primitive tools? On a clear night it is quite easy. All you need to do is plant a stick in the ground pointing straight up. Look down the length of the stick for a minute or so and one can see the stars directly overhead moving West. For daytime and cloudy nights how about a compass? I realize now that I was wrong about this. Magnetic fields are generated by geodynamoes. I was thinking that geodynamoes would be 4D vorticies but this is incorrect. As you may read in the thread on 4D vorticies there are two kinds, 4D self-contained eddies and 2D bathroom drain style vorticies that move stuff from one place to the other. (The vertical movement of stuff consumes one dimension so the vortex has only 3 remaining degrees of freedom and can't be a 4D vortex.) Geodynamos are of magma rising due to heat so they are of the 2D type. Their rotation would be in the 2-plane of the Coriolis force, the right-ana plane at any given location. A compass points in the direction of the dual of that plane, which is the up-east plane. A compass is designed to point only horizontally so it points to the East. Just like here on Earth it isn't exact but depends on local idiosyncrasies. Our Earth has three geodynamos, in regions where the Coriolis force is greater. On HyperEarth the Coriolis force is the same everywhere, so I’d expect more geodynamos.

So how to detect the other four cardinal directions? That's a puzzler, but I feel certain there is a way to distinguish them. It is analogous to a 2D person on a 2D planet. He knows up-down and forward-back but left-right is a mystery. How would I explain how to distinguish left from right? The math does not say anything about this. The assignment of + to right and - to left is an arbitrary convention that most people learn. We do it by asymmetries in our bodies. I can remember learning to do it when I was six years old. I had a callus on my right hand from using a pencil. I would feel that to distinguish left from right. In the vector algebra of physics the "right hand rule" is precisely this method. But this is of no help to 2D Man: he hasn't got a right hand, nor is there a right anything for that matter, so this is no aid to him. I'm going to assume that 4D people can tell right from ana due to some asymmetry of their bodies. It's too useful for them not to be able to do it. So the other cardinal directions would be right left ana kata while facing East. So let's just call them Right Left Ana and Kata. We pretty much do things this way, defining our east as "to the right when facing north." As for doing this more precisely I can’t think of a simple, primitive way.

Now suppose you are a traveler moving in one of these four directions. You will not be moving in the direction that minimizes the length of your journey. In other words, you are not moving in a geodesic. But this is not a problem at all. Indeed such a system arose naturally here on our own Earth with the Mercator projection. Its virtue was that it gave the compass bearing that would guide the ship to its destination. This is not the minimal distance and not a geodesic. It was also a rough approximation of what would actually happen. Currents and winds and so forth affected the ship. So they would set a course that was NOT directly toward the destination. If you set such a course and it didn't work then you had to guess which way to go. Guessing is bad. Instead they chose a course where you knew what the sense of sum of errors would be. Once you had travelled what you hoped was the proper distance you were sure you were to the northeast or whatever of where you wanted to go. Then you knew to travel southwest.

So a system that is an approximation is good enough for primitive times, as long as the direction of error is predictable. Pick the clock tower of Big City as your special point. Then the east west ana kata left right directions from there are good enough. Now suppose you want to go between two spots. If both are far from Big City the errors in simple math build up, but (I hope) you know the sense and rough magnitude of the errors are so that's OK. Once it starts getting out of hand you move to an artificial full-planet system based on geodesics developed by astronomers.
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### Re: Naming of 4D cardinal directions

Here are some names I have played around with for cardinal directions in higher dimensions, based on classical greek terms, trying not to repeat the same first letter:

North, South, East, West, Arctos, Boros, Dyst, Hesper, Ios, Oros, Pelia, Mast, Uris, Zephyr

northern, southern, eastern, western, arctern, borthen, dystern, hespern, iosern, orostern, pelien, mastern, uristern, zephyrn
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### Re: Naming of 4D cardinal directions

adam ∞ wrote:Here are some names I have played around with for cardinal directions in higher dimensions, based on classical greek terms, trying not to repeat the same first letter:

North, South, East, West, Arctos, Boros, Dyst, Hesper, Ios, Oros, Pelia, Mast, Uris, Zephyr

northern, southern, eastern, western, arctern, borthen, dystern, hespern, iosern, orostern, pelien, mastern, uristern, zephyrn

For 4D, another pair of terms ought to be sufficient. I personally like the sound of arctos/boros (even though I can't make sense of the Greek roots ).
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### Re: Naming of 4D cardinal directions

I like Arctos and Boros too.

When it goes to more than 4 dimensions I'll stick with prosaic numbers. Hinton used classical type names back there in the 19th century but I found this inpenetrable. I'm not willing to deal with that.
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### Re: Naming of 4D cardinal directions

PatrickPowers wrote:I like Arctos and Boros too.

When it goes to more than 4 dimensions I'll stick with prosaic numbers. Hinton used classical type names back there in the 19th century but I found this inpenetrable. I'm not willing to deal with that.

I think past a certain point, you need to have a systematic nomenclature rather than just inventing more arbitrary names. Like Wendy's Greek/Latin suffix/radix system in her Polygloss, from which predictable systematic names can be generated. Some kind of mapping from numerical dimension indices to lexical roots from which surface forms are derived.

E.g., the first few dimensions have ad hoc roots: 0D: -teel-; 1D -lat-; 2D: -hedr-; 3D: -chor-; but from 4D onwards the roots are regularly derived from Greek numerals: 4D: -tet-, 5D: -pent-, 6D: -hex-, and so on. All derivations are regular, though: teelon - point; latron - edge; hedron - polygon; choron - polyhedron. With different suffixes, different objects of specific dimension are indicated: hedrix = 2D manifold, chorix = 3D manifold, terix = 4D manifold, and so on.

In terms of cardinal directions, the first 2 pairs are already fixed, and ad hoc: north/south, east/west. The 3rd pair can probably be ad-hoc named arctos/boros. But probably starting from the 4th pair onwards, some kind of regular derivation would be desirable. Perhaps a regular suffix or regular mutation of root-initial consonant or something like that for deriving pairs of names from the sequence ... -tet-, -pent-, -hex-, etc.. Or perhaps, to prevent overloading the Greek numerical roots (when speaking of a sufficiently high dimension the Greek roots would dominate the lexical structure of the text, which can make it hard to parse), we could use a different sequence of roots taken from, say, Russian or something. Or something exotic like Austronesian, for maximum lexical distance from Greek roots.

Perhaps one direction could derive from Russian numerals and the corresponding opposite direction from Austronesian numerals, starting from 4 (because we already have north/south, east/west, and we agreed on arctos/boros). So you'd have two sequences: -chet(y)r-, -pyat-, -shest-, -sem-, -vos(e)m- (Ru); -sept-, -lim-, -enm-, -pit-, -wal- (Au) - with some vowels deleted to make every root monosyllabic. Supposing we arbitrarily pick -(o)th as the standard directional suffix, we'd have: chetroth/septoth, pyatoth/limth, shestoth/enmoth, semth/pitoth, vosmoth/walth as the next pairs of cardinal directions. (Vowel randomly deleted to make shorter words... but for consistency we could also always keep the vowel).
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