## 4-D wheels

Ideas involving the use of more than three spatial dimensions. If you want to talk about spacetime go to the Time Dimensions forum instead!

### 4-D wheels

whould anyone who is smart enouf to understand four-dimentional wheels share what they know with me. For example what is a spherindrical wheel? And how many other kinds of wheels are there?
arkmioh
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### Re: 4-D wheels

Try spellchecking next time.

Also: http://teamikaria.com/hddb/classic/page6.htm

Keiji

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### Re: 4-D wheels

Fortunately roads are flat whether they are in 2D space, 3D space or 4D space.
Though we may not think of a 4D road as flat; they would.

A 2D road is simply a straight line that undulates down and up through one direction hills.
A 3D road is simply a ribbon that undulates and turns left and right.
So to go from 2D to 3D we've just projected into the extra dimension.

The same goes for a 4D road.
You need to just take 3D ribbon and project it into an extra dimension and you now have your 4D road.

The 3D road is an infinite series of 2D roads side by side. A 4D road is an infinite series of 3D roads kata-side by kata-side.

Now the primary function of a wheel is to roll on the road and to hug the road.
A 2D wheel need only be a circle.
A 3D wheel need only be a cylinder; which again is an infinite series of 2D wheels x width.
A 4D wheel need only be a hyper-cylinder; which is an infinite series of 3D wheels x kata-width. I imagine that width would equal kata-width.

These 4D wheels will then nicely hug the 4D road while successfully driving over them.

The road would maintain its ribbon stretched into the extra dimension shape for its length but,
the road would be interesting in that it would twist as well as changing directions; whereas ours just go left and right.

The axles would also be interesting in that they need to pivot through 3 dimensions instead of 2.
gonegahgah
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### Re: 4-D wheels

Actually, that's wrong...
True a 2D road forms a line going up and over hills.
True a 3D road is the same road projected sideways into a ribbon with added turning left and right.

The 3D road can only project right because that is the only direction left.
If has to maintain its line like direction and it has to stay flat so it can't project upwards except to follow the top of hills and bottom of valleys.

Also if a 2Der were to to try to draw a road they would have to represent the turns on their vertical.
They would also have to represent our turns as shading as they represent all their corners.
The problem is that they lose vertical elavation because they can only depict our sideways with their 2 dimensions.
The answer for them is to use colour to represent elevation.

So as I said already the 2D road can only be projected in a line sideways to produce a 3D road.
Sideways in 4D is not a line.

The true shape of a 4D road would be a cylinder that wanders as a cylindrical line through the 3 sideways dimensions and over hills in the up dimension.
Now, just as for the 2Der, if we try to depict a 4D road we need to draw a cylinder travelling freely through our 3D space; which loses the up dimension.
Trying to project this up/down directly will generally look just like the cylinder moving through its sideways dimensions.

There are a couple of tricks we could use I suppose.
We could make the cylinder grow in circumference as it gets higher and shrink in circumference as it gets lower.
We could use the colours of the rainbow to depict elevation with purple as the lowest points and red as the highest points.
We also have to consciously remember that we aren't seeing the bulk which is lost in our 3D representation.

If we only looked at a 0 high elevation cross section what we would probably see is just a joining of two winding cones joined at the base.
These would start of as points and peter off to points as they dissappear finally into the elevation above or below.

Now, I'm still not sure how to put a wheel onto that hyper-surface so I'll have to work that out further in my brain...
gonegahgah
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### Re: 4-D wheels

Maybe you could use a psuedo-shadow to depict height above the lowest point?
Can you imagine the roads. They would probably happily loop through and around themselves as they climb or descend the 4D hills and valleys.
gonegahgah
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### Re: 4-D wheels

The penny slowly drops. I must have dropped it in treacle?

I just realised that only the tyre track would reside on 'top' of the road pretty much as quickfur has indicated also.
By a stretch of superimposition we might draw the tyre track inside the road but it would actually dimensionally be just infinitesimally above it (unless the road is soft).
But, ignoring this, as for the cylinder road I would consider that the tyre track would also be a cylinder as their is no preferred left-right over ana-kata.

So, if you left tyre tracks behind they would be a set of long cylinders thinner than the road and that parallel each other whilst travelling the length of the road 'through' it.
I'm not sure how many tyre tracks there would be at this stage.
In 2D there is one tyre track; in 3D there generally need be only two; and possibly in 4D there may need to be three to give balance?
gonegahgah
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### Re: 4-D wheels

Thinking about it I have to presume that you would need three legs for balance in a 4D world.
It would also suggest that a 2Der would only have one leg as they do need the second leg for sideways balance.

For a 4Der the 3 legs would not get in the road as they are all side by side to each other.
Very unlike a tripod creature in our world where the third balancing leg sits in the direction of travel making walking ungainly.

In our world, and all these worlds, we need a set of wheels at the front and at the back so that our vehicle can be long and not tip forwards or backwards.
So it stands to reason that a 4D car will have six wheels as standard.
gonegahgah
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### Re: 4-D wheels

I am now thinking that the progression for sidedness would be like the following:
1Der has 0D sidedness which is represented by a dot = no sidedness
2Der has 1D sidedness which is represented by a line = left side and right side
3Der has 2D sidedness which is represented by a full circle = 360deg of sidedness.
I've been now been puzzling about the sidedness and this is what I have come up with...
gonegahgah
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### Re: 4-D wheels

Maybe you would also need to have three tyres per wheel spot or a total of nine at the front and nine at the back?
Apart from the need to 'climb' forward it would help towards the extra traction needed for the extra momentum of 4D.

Again the tyre tracks would look like cylinders aligned in the direction of travel and they would be spaced as the points in an equilateral triangle.
Mags would leave bigger cylinder tracks.
Of course the tyres would each extend into the upper dimension, that I am hiding, as well.

The trio of wheels would be spaced around like three small equally spaced fractional parts a donut into the upper dimension.
The wheel steering arms would pivot the trio of wheels all as one towards any of the 360deg sideways; whereas our 3D wheels only pivot along a line.
This would provide the necessary steering to get to any part of the 4D road; which we are seeing as a cylinder aligned to the direction of travel.

How does that sound?
gonegahgah
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### Re: 4-D wheels

You can generally, work out the general shape of a wheel in N dimensions, they are pretty much the same: a spherated circle.

You could think in terms of say, Thor from the B.C. cartoon. The wheel consists of an axle protruding on each side, and one stands over the wheel with a foot on each side.

What ever figure we take, we now divide it by time (forward) and gravity (height). Any other space left over is 'across'. In terms of this, there is a central point 'straight forward', and one can move to any point away from the centre makes a turn to that direction. The distance or duration of the awayness gives the intensity of the turn, like in a motor-car. You turn the steering wheel left, and it turns left, but the harder or longer you hold left, the more angle it goes through.

To effect this kind of steering, the helm has to be a sphere in the across-space, ie a ring. One then in the simplest move, pulls or presses the ring in the direction to be turned into: Thor would press on the left axle to turn left, and the right axle to turn right. So the steering wheel in 4d, would facilitate a full across-space, prehaps in the sense of a sphere, which one turns to turn into the direction to be turned.

What ye have in 4d, is that the across-space has several dimensions, and it is possible to set a rotation up in this space. This is not good for driving in since one would be effectively 'rolling', to use the aeroplane term.
The dream you dream alone is only a dream
the dream we dream together is reality.

wendy
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### Re: 4-D wheels

I'll have to press my brain and conceptualise the tyre shape still. Haven't done that yet.
I was thinking today about the rotation aspect you mention though and how it would be controlled.
Ideally as you round a corner you would want to have your rotation aligned to making the shortest route.

The other interesting thing is that two cars can pass you at once just on your inside track (or just outside if you are going slower) in 4D.
It might also mean that there are more cars in a 4D race?

Certainly you don't have to just pass other cars on just two sides but can do so on many sides within the 360deg of sidedness on the straights.
I'm not sure how to add rotation to driving but maybe you would need 3 keys to steer in 4D on a race track steer to simulate how they would think?
We steer with 2 keys in 3D. As I say I don't know how rotation would be added and removed?
gonegahgah
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### Re: 4-D wheels

A 3D tyre is a cylinder with a cylinder space in the middle.
This can be considered to be an infinite series of concentric empty cylinders as per the following picture:

I would think a 4D tyre would be an infinite series of concentric empty cylinders just like a 3D wheel...
...except the concentric cylinders would not have flat walls...
...they would have circle walls so that to us each concentric cylinder would look like a donut.
Each concentric donut would grow smaller towards the centre but a cross section of the donut sides would stay the same radius;
just like the 3D tyre above keeps the same width as the concentric circles get smaller.

For the moment I've describe the tyres as solid whereas they are primarily empty to hold air.
I just wanted to describe the overall appearing shape. I'll try to get a friend to draw it for me.

You have to remember that this is in 4D so the infinite circles that form a donut are actually all literally side by side at their faces.
The inside of these donuts is actually the same size as the outside of that donut;
unlike in our world where the inside circle of a donut is smaller than the outside circle.

How does that sound? And what shape have I described? Is what I'm describing a spherated circle or something else?
gonegahgah
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### Re: 4-D wheels

Should clarify the donut inside business. When I provide the next picture it should be remembered that I'm trying to show all 4 dimensions at once.
Although each concentric part of the wheel will look donut like; in fact the sides - which are all directions of the circle cross section - are actually all in the side dimensions.
I'm trying to depict up, forward and two side dimensions.
Maybe I can skew the wheel sides - which cover a whole 360deg - to show this sidedness better?
gonegahgah
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### Re: 4-D wheels

Roads are linear in all dimensions, since they have to start somewhere and end up somewhere else. If they weren't linear, they wouldn't be roads anymore (you could have a baseball field between you and your destination, but the path you'd take would still be linear: you can't move in more than one direction at a time). However, the degree of freedom in which they may curve differs from dimension to dimension, and this also determines the complexity of road systems that may be formed.

In 2D, roads take up the entire terrain, and there's only forwards and backwards. There are no lateral dimensions to curve in, so all roads are straight (save perhaps for a bit of uphill/downhill slope). Where there's a road, there can't be buildings or any sort of structure; the surface of the planet is essentially a bunch of buildings with roads in between, and wilderness outside.

In 3D, roads do not take up the entire terrain, but they do divide the surface of the ground into two parts on either side. There's 1 lateral dimension, so roads can zigzag left and right. They also may slope uphill/downhill. When there are multiple roads in different directions, they generally must intersect with each other. So junctions are very prominent in 3D road systems, with elaborate traffic control systems that eventually develop, such as traffic lights, right-of-way rules, left/right lane directions, and so on. In cities, roads divide the city into blocks.

In 4D, roads not only do not take up the entire terrain, they no longer divide the surface of the ground. Roads in 4D have only a single "side" that wraps around the road. There are 2 lateral dimensions, so roads not only can zigzag, they can spiral. This is in addition to the usual uphill/downhill slopes, of course. Quite unlike 3D, when there are multple nearby roads going in different directions, they need not intersect. In fact, two straight roads in 4D probably will never intersect, even if they pass close to each other. Because of this, junctions are not prominent in 4D road systems at all. Rather, it's much more likely to have a system of on-ramps and off-ramps, like we have with highways/freeways (because roads never need to intersect). So there is no need of traffic lights at all, just a system for splitting off the off-ramps and merging from on-ramps. Except that in 4D, these aren't ramps at all; they are flat on the ground.

Furthermore, in 4D cities, roads do not divide the city into blocks at all. In fact, it's entirely possible for a single building to "surround" a road in a way that it never intersects the road, yet it is fully connected without any overhead bridges/walkways.

The purpose of a wheel, at least as far as a vehicle is concerned, is to roll forwards/backwards in a single line, thereby transporting a payload with minimal energy expenditure. To that end, the wheels of the various dimensions are generally quite simple. In 2D, it's just a circle (a 2D "ball") that rolls along the road. In 3D, it's still just a circle that rolls along the road, except that to keep itself upright, it needs to extend into the 3rd dimension, so you end up with a cylindrical (or perhaps torus) shape.

Similarly, in 4D, a wheel is still just a circle... except that now it needs to extend in two dimensions in order to be able to support itself. So you end up with the Cartesian product of a circle and a square (that is to say, a circle extruded twice), which is a duocylinder. Which can also be obtained by extruding a 3D cylinder in the 4th direction.

Of course, this doesn't fully describe the class of shapes that may roll in one or more directions. In 2D, the circle is the only rolling shape. In 3D, the cylinder can roll, but so can the sphere. Unlike the cylinder, which only rolls in a single line, a sphere can roll in two directions simultaneously, so it can cover a 2D area by rolling. (Think of a billiard ball: it can roll anywhere over the surface of the table.) The sphere is not very useful as a wheel, though, because its ability to roll in many directions, while cool, is difficult to control. It's possible, of course, just not very practical.

In 4D, you have the cubinder, which is the best shape for wheels. You also have a spherinder (the extrusion of a sphere; IOW a 4D cylinder with two spheres as "lids"), which can roll in two directions at once, like a sphere. It can therefore cover the space of a 2D region just by rolling. However, in 4D, this is quite constricted, because the ground is 3D, so the spherinder can only cover a small part of the ground. They are difficult to use as wheels for the same reason as a sphere in 3D, but there may be some applications for them that take advantage of the fact that they are confined to roll along a 2D subset of the ground. There is also the 4D sphere, which can roll in 3 directions simultaneously. It's basically the 4D equivalent of a ball, so it's useful as a sports object, but difficult to use as a wheel.

In 4D, there's also a new kind of rolling object with no equivalent in lower dimensions: this is the duocylinder. It has a peculiar property that it consists of two surfaces in a torus shape. Unlike the spherinder, which won't roll if you stand it up on its end, the duocylinder can always roll no matter which side you stand it on. However, the direction it rolls when stood on one side is perpendicular to the direction it will roll when you push it over to rest on its other side. The duocylinder is not easy to use as a wheel, because its peculiar shape causes it to always be able to roll on its other side when you confine it on one side. However, it may have some uses in conveyor belts, where it allows free movement along one axis, but always pushes objects resting on it along a perpendicular axis. It may also find use in mechanical devices, though further study needs to be made in this area.
quickfur
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### Re: 4-D wheels

That certainly makes sense. No need for traffic lights! That's very cool. That helps to build towards the bigger picture. Thanks quickfur.

When I get the tyre drawn that I'm describing and put it here I'll be curious what type of shape it is. I look forward to getting your knowledge on that.
Just from the description of a cubinder I think that what I'm describing is slightly different.

I'm thinking that a 4D person won't think of sideways as having two directions. I'm thinking they will see it as a continuity of the full 360deg.
I'm also thinking that 4D space and physics won't tend to treat them as two separate directions of sideways either.

So, whereas the cubinder takes the forward rolling 2D circle, extends it one sideways, then extends it the other sideways,
I'm tending to think that what 4D will do is take the forward rolling 2D circle and extrude it all sideways directions (360deg) equally at once.
This will create a circle into sideways centred on the original rolling 2D circle.

Our 3D tyres do the same thing. They extend into all available directions of sideways equally (but there is only 1 sideways).
In 4D there is a whole 360deg of sideways so I'm guessing that 4D nature won't differentiate.
You'd most likely get even more bizarre shapes of crystals for example as you do even in our 3D world that completely ignore our ideas of things shaping in 3 directions only.

I'm just thinking that a cubinder would be similar to us creating our wheels like iceskates?

Does that make sense? I look forward to the name of the shape I'm describing.
gonegahgah
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### Re: 4-D wheels

With the shape I'm describing, if you were to squash all the 'up' dimension together into a flat pancake and have the forward dimension in one direction and the 360deg of side dimension perpendicular to this following through the vacated up direction around the other available sides around to the bottom, the resultant shape would look like a cylinder where the direction of roll is the front flat end of the cylinder. ie.:

If we were to look at a series of 'up' cross sections then we would get the following:

That is assuming that the tyre didn't squash at all which it would. The squashed parts would be cylinders.
This also ignores that the tyre has thickness and is treating it as if we would treat a 3D tyre as a round ribbon which it isn't.
Adding thickness would provide extra inner circles that follow the same pattern except describing a smaller circle through the vertical.
Though slice-wise they may distort through the height just as a 3D wheel sliced vertically presents different cross-sections.
gonegahgah
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### Re: 4-D wheels

To get a 4D wheel you could probably just take every up slice cross section of our 3D wheels, take that slice and rotate it the whole 360deg through the total side dimension to give us the up cross-sections of a 4D wheel.
gonegahgah
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### Re: 4-D wheels

gonegahgah wrote:[...]I'm thinking that a 4D person won't think of sideways as having two directions. I'm thinking they will see it as a continuity of the full 360deg.
I'm also thinking that 4D space and physics won't tend to treat them as two separate directions of sideways either.

The thing is, the division of lateral direction into forwards/backwards and left/right is arbitrary.

In 3D, you can indicate horizontal direction by a circle that lies horizontal, with a red dot marked along some point, indicating the direction you're going towards. By rotating the circle, the red dot moves around in a full 360°, so that's all the directions that you can move along the ground. So forwards/backwards is really just an artifact of how the red dot lines up with your body's orientation. If the red dot is in front of you, then the direction is forwards; if it's behind you, it's backwards. If it's next to your left hand, then it's a leftwards direction, and if it's next to your right hand, it's a rightwards direction. But as far as horizontal direction itself is concerned, there is no distinction between forwards/backwards and left/right. It's a full 360° freedom.

In 4D, the equivalent is a sphere with a red dot marked. The sphere has full 3D rotational freedom, so it can point in any of the 6 cardinal directions in 3D plus everything in-between. Remember that in 4D, all of these directions are horizontal. Again, there is no distinction between forwards/backwards and sideways (which in this case is a 2D sideways). The fun part comes with how these directions relate to a 4D being's body.

To simplify things, we can make the very reasonable assumption that a 4D being will be facing some specific horizontal direction (conceivably a 4D being can face multiple directions at once, but that just needlessly complicates things, just like trying to understand relative directions from a hydra's point of view in 3D). If the red dot on the sphere is in front of the 4D being, then, it's regarded as a forward direction; if it's behind, then it's backwards. That leaves two more dimensions laterally. So we need additional reference points here. For no good reason, let's assume that the 4D being's upper limbs lie at a point perpendicular to its forward/backward direction (just like our arms don't grow from our chest or back, but from our sides). So this constricts the 4D being's lateral limbs to the hyperplane that lies perpendicular to its front/back.

However, this being 4D, that still leaves 1 whole dimension left: so the 4D being can still have a circle of arms all around its body, yet they are all simultaneously perpendicular to its front/back! (One way to think of this is that the cross section of a 4D being's body with a hyperplane is a sphere, with the north pole being "front" and south pole being "back", so that leaves the entire equator for arm placement.) So where we go from here depends on another very important assumption: does the 4D being have circular symmetry about its lateral sides, or does its body shape have some preferred direction?

For example, if the 4D being has two arms on opposite sides of its body, that amounts to marking some arbitrary point on the equator of a sphere as the "absolute east" and the opposite point as the "absolute west". In this case, then, if the red dot of the steering sphere lies close to one of these points, it would be regarded as a "left/right" direction, and if it's not close to any of these points, and is neither forwards/backwards, then it would be ... uh... marp/garp? ana/kata? But it would be perceived differently.

On the other hand, if the 4D being has, say, 4 arms arranged in a circle at 90° with each other, then it's more likely to consider the entire 360° of lateral directions as being approximately equivalent, so it wouldn't give special significance to any particular lateral axis. So "sideways" could mean any point along a 360° sweep. (Of course, the 4D being would probably still be biased towards where its arms are, it would use them as points of reference. So it'd refer to left/right arms or ana/kata arms, even if the two are equivalent via a 90° rotation.)

So, whereas the cubinder takes the forward rolling 2D circle, extends it one sideways, then extends it the other sideways,
I'm tending to think that what 4D will do is take the forward rolling 2D circle and extrude it all sideways directions (360deg) equally at once.
This will create a circle into sideways centred on the original rolling 2D circle.

What you describe is a duocylinder.

Our 3D tyres do the same thing. They extend into all available directions of sideways equally (but there is only 1 sideways).
In 4D there is a whole 360deg of sideways so I'm guessing that 4D nature won't differentiate.

This is true. From a 4D being's POV, there's no reason to be constricted to a cubinder for a wheel. It can be any n-prismic cylinder. For example, it could be a hexagon+circle prism, in which case the point of contact with the ground would be a hexagon. The duocylinder is just the limit of n-gon+circle prisms, so it could work as a wheel too.

Except that a duocylindrical wheel has an unconstricted rotation in the axis parallel to the ground, so your car could be spinning wildly around a lateral direction but yet be upright and still facing the same forward direction the whole time. Not a problem for getting from point A to point B, but might be very confusing for the driver. Some kind of non-circular lateral cross-section is desirable in order to keep the car in a fixed lateral orientation.

In fact, there's no reason the lateral cross-section of a 4D wheel needs to be a regular shape at all. Any kind of toroidal shape in the vertical plane would be good enough for forward/backward movement, so in theory you can have a hexagon-star+circle prism for a wheel, where the point of contact with the ground is in the shape of a hexagonal star. You can have any arbitrary 2D shape here; the Cartesian product with a circle ensures that it can always rotate when upright.

You'd most likely get even more bizarre shapes of crystals for example as you do even in our 3D world that completely ignore our ideas of things shaping in 3 directions only. [...]

Crystal shapes are dictated by the efficiencies of sphere packings. I don't know enough about sphere packings in 4D to say for sure, but I would expect the 24-cell (and derived shapes with 24-cell symmetry) to feature prominently in 4D crystal shapes. Perhaps a bi-24-cell (the convex hull of a 24-cell union its dual -- which if I'm not mistaken is one of the uniform duals).
quickfur
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### Re: 4-D wheels

You're right, I was starting to realise that perhaps the shape could be arbitrary too but I am trying to consider if one shape is more desirable than another for any reason.
Our 3D tyres needn't be perfect cylinders either. They could have wave like shapes. ie.:

This wheel is always flat to the ground as it turns but it is probably not practical to make wheels like this.
So perhaps there are other practical considerations such as having too many right angles that would perhaps weaken the tyre?

I also imagine in a 4D world that you will want rotation as well as direction. When we shoot upwards into space we may rotate however we like as we go.
Obviously the more spin you get the greater the sideways forces but it may allow our 4D race cars to slip around those corners just that degree tighter to the corner.
In general we tend to make our rockets round as well as they have no preferential side direction. The shuttles aren't cylinders as they have to land.

I do think that there needs to be a mechanism to add and remove rotation. I don't think the car would spin of itself like this though corners may somehow add some spin.
I think the steering wheel could be designed to handle both turning as well as rotation in a 4D world.

Why would a cubinder prevent spin? You could spin a boxy shaped rocket just as well as a cylindrical one.
I'm not sure if a duocylinder is the shape I'm describing or not yet???

As mentioned in the other threads I'm tending to believe that 4D creatures will have three of things. Three legs, three arms, two claw-posable thumbs.
So their thinking would perhaps be more tri-side direction than quad-directional. I'm suspecting they would also have to have three eyes to determine distances along two axis.
gonegahgah
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### Re: 4-D wheels

gonegahgah wrote:You're right, I was starting to realise that perhaps the shape could be arbitrary too but I am trying to consider if one shape is more desirable than another for any reason.

I suspect a duocylinder might actually be more desirable than some arbitrary shape, like a hexagon, unless there's a specific reason for it. Such as threading the hexagonal cross-section through a hexagonal-prism hole of some sort to maintain/control orientation.

Our 3D tyres needn't be perfect cylinders either. They could have wave like shapes. ie.:

This wheel is always flat to the ground as it turns but it is probably not practical to make wheels like this.
So perhaps there are other practical considerations such as having too many right angles that would perhaps weaken the tyre?

Assuming we're using inflatable tyres, it would make sense for the shape to naturally assume a duocylindrical configuration, under the pressure of compressed air inside it.

I also imagine in a 4D world that you will want rotation as well as direction.

True; for that having a non-circular cross-section, such as a hexagon, makes it easier to build control mechanisms for orientation. OTOH, the presence of an axle would prevent rotation in the plane parallel to the ground, so perhaps this is a moot point. Changing orientation in that plane would require some sort of pivoting mechanism to rotate the body of the vehicle around (but without changing its forward direction or vertical orientation -- that extra degree of freedom in 4D is both weird and cool at the same time).

When we shoot upwards into space we may rotate however we like as we go.
Obviously the more spin you get the greater the sideways forces but it may allow our 4D race cars to slip around those corners just that degree tighter to the corner.
In general we tend to make our rockets round as well as they have no preferential side direction. The shuttles aren't cylinders as they have to land.

Spacecraft in general have rotation perpendicular to the direction of motion: the reason for this is that the resulting angular momentum stabilizes orientation (like how a fly-wheel resists changes in orientation). A 4D vehicle has full contact with the ground, and needs no such stabilization: its wheels keep it in place and in a stable orientation.

I do think that there needs to be a mechanism to add and remove rotation. I don't think the car would spin of itself like this though corners may somehow add some spin.

Actually, now that I think about it more carefully, I think spinning of itself would not happen unless the vehicle had all wheels in a single line (like a bicycle). Introducing an axle constrains the possible wheel rotations to 3 dimensions (so it can roll like a sphere), and the duocylindrical shape of the wheels constrain it further to two dimensions (only rotates in a single plane). The additional dimension leftover is not free to rotate -- you need at least two free dimensions for rotation to happen. So basically, having duocylindrical wheels connected by axles eliminates spinning motion. You can still make it happen by synchronously turning your wheels, so that the vehicle moves forward with a spiralling motion.

I think the steering wheel could be designed to handle both turning as well as rotation in a 4D world.

It'd be a duocylindrical steering wheel!

Why would a cubinder prevent spin? You could spin a boxy shaped rocket just as well as a cylindrical one.

You're right, it can still spin since the point of contact with the ground is only a square, as long as you're also moving forward, this will permit spinning.

I'm not sure if a duocylinder is the shape I'm describing or not yet???

I'm pretty certain it is. It wouldn't be the usual duocylinder with two equal radii, of course, but more likely one with a larger radius for one torus and a much smaller radius for the other.

As mentioned in the other threads I'm tending to believe that 4D creatures will have three of things. Three legs, three arms, two claw-posable thumbs.
So their thinking would perhaps be more tri-side direction than quad-directional. I'm suspecting they would also have to have three eyes to determine distances along two axis.

I don't know about this, the arrangement of limbs in organisms doesn't always follow "sensible" patterns. In 3D, for example, the most stable standing structure is one with three legs. However, no animal has three legs (none that I know of anyway): they either have four, which is redundant, or two, which is unstable (but somehow we manage just fine with only two). Insects have six, and arachnids have eight. Caterpillars have a whole bunch, for no apparent reason. Bilateral symmetry seems very popular among animals, but you do have things like starfish (and some types of plants) with pentagonal symmetry. So I can't see much justification for saying that in 4D, the number three is somehow preferred.
quickfur
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### Re: 4-D wheels

It's just a suspicion at this stage but I do have an initial reason why I think that 4Ders might have an extra leg than us.

In 4D you could really have any number of legs side by side without any of them getting in the way; though there is probably a limit due to available space :/
But I think minimalisation would tend to be the trend for higher order land based vertebrates hence we have two legs (though not one because hopping is not a good look).
I guess less legs means that we can dedicate less brain power to co-ordinating our walking I guess; maybe.

The other aspect of our having two legs (and another reason for the lack of one legged hoppers) is to allow us to move forward freely while maintaining a steady sideways balance. We can fall gracefully foward to walk without having to worry about falling sideways. Having two legs not only provides us with another leg to walk to but also provides us with stabilisation hence why they are side by side and not one in front and one behind (and also so that the strides can be longer).

However, a 4Der can fall sideways in 360deg of direction rather than just left or right.
And a tripod as you also mentioned is certainly the minimum that you need to prevent falling in 360deg of direction.
So as a minimum I suspect that we need at least three legs, to allow the sort of stable walking that we take for granted, to occur in 4D.

That's my reasoning for it anyway?

If it is so then whereas we tend to have a sidewards pendulum gait they would tend to have a more triangular/circular gait as they walk.
gonegahgah
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### Re: 4-D wheels

gonegahgah wrote:It's just a suspicion at this stage but I do have an initial reason why I think that 4Ders might have an extra leg than us.

Yes, I didn't say that 4Der's must have two legs. In fact, evidence suggests that they should have at least three, probably more.

In 4D you could really have any number of legs side by side without any of them getting in the way; though there is probably a limit due to available space :/
But I think minimalisation would tend to be the trend for higher order land based vertebrates hence we have two legs (though not one because hopping is not a good look).

I don't think good looks is a real criterion in deciding body plan. Well, except for us, maybe, since we're just speculating.

It's entirely possible to have a single leg... just that it would be a major pain to stand up, and a challenge to keep balance, because you have a single limb to balance in 2 dimensions (try standing on one leg for long periods of time, for example). In 4D, a standing creature would have to balance in 3 dimensions (there are 3 ways in which it can fall). Having at least two legs reduces it to 2 dimensions, but that's still only the equivalent of us 3D humans standing on one leg: balancing is hard. So I'd expect a 4D creature to have at least 3 legs, even just for balancing purposes. I would expect 4 legs should be more common, because an even number of legs makes it easier to have a balanced walking gait (two legs to hold current position while the other two move forward, and vice versa).

In fact, I've speculated in the past about 4D creatures with a tetrahedral arrangement of legs. Their walking gait would consist of holding two legs in place while moving the other two forward, with alternating pairs of legs. In projection, the movement of their feet would be like a tetrahedron that inverts itself along the line that bisects two of its edges. (In the middle of the gait the arrangement of legs would be in a square configuration.) This arrangement also has the advantage that when standing still, they actually have maximal stability (4 legs in tetrahedral formation is the most stable standing configuration in 4D).

I guess less legs means that we can dedicate less brain power to co-ordinating our walking I guess; maybe.

Nah, once learnt, walking is 99% muscle memory. If you think too hard about how to move your legs when you walk, you'll suddenly discover that you don't know how to walk. Normally we don't think about it, we just do it.

The other aspect of our having two legs (and another reason for the lack of one legged hoppers) is to allow us to move forward freely while maintaining a steady sideways balance. We can fall gracefully foward to walk without having to worry about falling sideways. Having two legs not only provides us with another leg to walk to but also provides us with stabilisation hence why they are side by side and not one in front and one behind (and also so that the strides can be longer).

However, a 4Der can fall sideways in 360deg of direction rather than just left or right.
And a tripod as you also mentioned is certainly the minimum that you need to prevent falling in 360deg of direction.
So as a minimum I suspect that we need at least three legs, to allow the sort of stable walking that we take for granted, to occur in 4D.

Yeah, at least 3 legs are necessary for balance. For walking, I postulate 4 as the next smallest even number, so pairs of legs can alternate in the walking gait.

[...]If it is so then whereas we tend to have a sidewards pendulum gait they would tend to have a more triangular/circular gait as they walk.

Using the 4-legged tetrahedral arrangement I described above, the walking gait would alternately swing in one of the two horizontal axes perpendicular to the direction of motion. So if you call the vertical direction W, the forward direction X, then the walking creature would alternately swing along the Y axis, then the Z axis, then the Y axis, etc..

A more extreme scenario I thought of is 6 or 8 legs in a circular formation. In this case, there is no stabilization in the forward/backward direction, so we may assume that the feet are extended along the forward axis to provide minimal balance when standing still. When walking, the legs move in two groups. Each group consists of every other leg in the circular formation. So the 6-legged arrangement behaves like two dual triangles, and the 8-legged arrangement like two dual squares. This arrangement gives maximal sideways stability when walking (the triangular or square arrangement of the stationary group of legs fully support the creature in all sideways axes), so the creature needs no extra effort to balance itself sideways when walking.

The 8-legged creature can also adopt a significantly more stable stance by staggering each of its groups of 4 legs, thereby forming an octahedral arrangement of feet on the ground. It could adopt this stance, for example, when sparring in martial arts, say, so that it's not easily knocked over.
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### Re: 4-D wheels

quickfur wrote:Yeah, at least 3 legs are necessary for balance. For walking, I postulate 4 as the next smallest even number, so pairs of legs can alternate in the walking gait.

When they refer to 'muscle memory' I'm fairly certain that that memory still occupies the brain. The brain remembers particular triggering patterns. So I would still wonder if having more legs, which requires maybe more co-ordination, would consume more brain space in animals with more legs?

But anyhow, what I am still contending is that just because we are bi-pedal - and in general our animals are multiples of - I don't believe there is an imperative for higher dimensional creatures to have an even numbers of legs at their front (or back if they have back legs). We only need two legs for balance as we have two sideways directions to fall. So I contend that as a minimum they can have three legs when you have 360deg of sideways direction to fall. Two legs would be difficult to balance on for a 4D homo erectus but they could balance on three against falling sideways. A fourth leg at front (and/or back) is superfluous. I would expect the three legs to be at 120deg sideways to each other with each centimetre above ground all in the same sideways planes which is 90deg to the forwards direction. The feet would form a triangle with the ground leaving the forward direction of the ground free of any legs.

I don't think there is a need to go to four legs. The same for a 5D creature; I believe they can balance sideways on just four legs, A 6D creature on just five legs, etc.
Perhaps for the higher dimensional creatures the issue of bulk may necessitate more legs but they would also be much smaller in length than us so it's hard to tell.

In some respects we have feet to help with our forward balance without disrupting our sideways maneuverability too much.
If there weren't a need for sideways maneuverability we would probably have semi-spherical pods for our feet.
So our higher dimensional kindred are likely to have forward pointing feet too.
Last edited by gonegahgah on Thu May 17, 2012 12:12 pm, edited 2 times in total.
gonegahgah
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### Re: 4-D wheels

I've got a friend working on the 4D tyre so I will be curious to see if it matches exactly the duocylinder. The drawing won't look exactly like your depiction quikfur. On the projection, we are doing what looks like the outside of the donut (which is actually equally the side) connects to the inner torus of the donut on the opposite side. And vice versa. So it sort of looks like a donut that passes through itself from both sides. So I am curious if they are the same thing.

The other thing is the steering mechanism. Our cars use a pin and a steering arm. The steering arm is fixed to the wheel and the wheel is pinned to the car and the steering arm is pinned to the steering rack which the steering column moves left or right to push/pull the wheels around their fixture point to the car.

A 4D wheel needs to tilt (doing so like ours about 45deg either way) not just backwards and forwards through two directions of sideways but needs to tilt backwards and forwards through every angle of the whole 360deg of sideways.

... Sorry, I have a puppy who I'm keeping an eye on to ensure they be good so I'll get back to thinking about this later...
gonegahgah
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### Re: 4-D wheels

... and so ....

If you have the 4D wheel affixed to the 4D car via a fixed pin 'inside' the centre of the wheel then the closest analogy we can give in our 3D world is probably:

Basically there is 360deg of tilt for the wheel but it all is towards the forwards direction or backwards direction.
I've depicted forwards in all directions that has to be squeezed back to one direction for 4D.

The other important thing is that we 3Ders can go forwards to the left and forwards to the right.
They can go forwards towards 360deg of sidewayness.
And we can go backward steering towards left or right.
The same goes for our 4D creatures who can go foward towards 360deg of sidewardness and backwards towards 360deg of sidewardness.

I've tried to depict this in my diagram with 'back-sideways'.
Tilting the wheel the opposite angle of sidewayness is not the same as going backwards as can be seen in our 3D world steering.
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