4d Helicopters

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.

4d Helicopters

Postby anderscolingustafson » Thu Apr 10, 2014 9:25 pm

I was thinking about how a 4d Helicopter blade would work in 4d and I was thinking and I came up with a kind of cylindrical blade for a 4d Helicopter.

4d chopper.png
4d chopper.png (6.01 KiB) Viewed 14534 times


The green part represents the place in which the blade would connect to the Helicopter. In order to fly the cylindrical blade would spin around the axis in which it connects to the Helicopter to cut the equivalent of air.
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Re: 4d Helicopters

Postby Keiji » Thu Apr 10, 2014 9:42 pm

Trouble is, in 4D, you don't spin something around an axis, it spins "around" (or, rather, in) a plane. The connection to the helicopter would therefore have to be not a linear axle, but a planar one. But then, the helicopter would struggle to orient itself perpendicular to that plane, perhaps. Unless it had an extra rotor, like we have the tail rotor in 3D.
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Re: 4d Helicopters

Postby anderscolingustafson » Thu Apr 10, 2014 9:58 pm

The blade of a Helicopter is the n-1 dimensions so in 3d the blade is a 2d plane and so spins around a point. In 4d a Helicopter blade would need to be 3d to cut space and so even though a 4d object would spin around a plane a 4d Helicopter blade would need to be a 3d hyperplane as 4-1=3 so a 4d Helicopter blade would spin around an axis as it would be 4-1 dimensions.
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Re: 4d Helicopters

Postby Keiji » Fri Apr 11, 2014 5:41 am

Sorry, that's categorically incorrect. You cannot spin something around an axis in 4D. In any dimension, things rotate in a plane, so in 4D, they rotate in two dimensions leaving the other two stationary.

If the blade is 3D then the "axis" would have to be a 2D figure, not a linear figure.
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Re: 4d Helicopters

Postby quickfur » Fri Apr 11, 2014 2:41 pm

A 3D helicopter with only 1 rotor can't maintain a fixed pointing direction, because the single rotor fixes the vertical orientation but leaves 2 other dimensions left over, which is an extra degree of freedom. That's why a tail rotor is needed.

A 4D helicopter with only 1 rotor will also have a hard time maintaining its orientation, because a single rotor fixes the vertical orientation, but that leaves 3 remaining dimensions free to rotate, which is 2 degrees of freedom. So you'd need at least two equivalents of the tail rotor, at 90° to each other, in order to maintain a fixed orientation (although with a single rotor you will be able to maintain a fixed pointing direction, but the fuselage will still be able to freely rotate laterally, which will give the passengers a nasty headache and confound the pilots 'cos any attempts to turn the aircraft will cause it to spiral through the sky).
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Re: 4d Helicopters

Postby anderscolingustafson » Fri Apr 11, 2014 4:58 pm

In 4d there would be 6 different non vertical perpendicular ways something could spin and 3 of those six ways would be opposites of the other 3. So each of the 3 rotors would need to spin in a way that would be perpendicular to the other two in order for their spins to balance each other out.
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Re: 4d Helicopters

Postby quickfur » Fri Apr 11, 2014 5:06 pm

Yes, that's right.
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Re: 4d Helicopters

Postby anderscolingustafson » Fri Apr 11, 2014 7:47 pm

This gives me an idea for 4d bullets. I was thinking that in 4d perhaps a bullet could have 3 rotors behind it that each spin in three perpendicular ways in order to help propel the bullet forward and keep the bullet moving in a straight line.
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Re: 4d Helicopters

Postby quickfur » Fri Apr 11, 2014 8:02 pm

Rotors on a bullet??! Surely you mean guided missile? This is way overkill for a mere bullet.

Besides, I already said in another thread that the bicircular tegum (Cartesian product of a circle with smaller circle) is probably the best shape for a bullet, since it will maintain a sharp edge against its direction of travel regardless of how it may rotate laterally. Sorta like a cross between a bullet and a shuriken, in a way only 4D can. :lol: You just give its major axis a spin upon exit from the barrel, and the angular momentum should keep that plane fixed, and the orthogonal plane is free to rotate, which is no problem since the shape is symmetric about any rotation in that plane.
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Re: 4d Helicopters

Postby anderscolingustafson » Sat Apr 12, 2014 3:32 am

I wonder how a 4d fan would work.
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Re: 4d Helicopters

Postby wendy » Sat Apr 12, 2014 10:18 am

One needs to address how things like screws and eddies might work in four dimensions. These need to be answered before one starts writing fancy things like wave equations etc.

Anders suggests a cylinder-blade for the helicopter. I am not sure if this can work either separately or in multiple. Let's look at this.

A helicopter is an air-screw. The rotor at the back, serves to stop the cabin rotating. In its absence, the engine causes rotation on the main shaft, and this causes an equal spread of energy of the cabin spinning one way and the rotor going the other way. You have a rotor at the back to stop the cabin rotating. You can do this with several engines too.

Wheels for carrying things would rotate in the height-forward hedrix. Anything extra is 'across', and any extra rotations would be felt in the cabin. So ideally car wheels etc, do not want rotation other than height-forward. On the other hand, a planet does not see height-forward, but two rotation-modes against which energy must be equalised. That's why the 4d earth spins in clifford-motion, while the wheels turn in great-arrow manner (keeping N-2 dimensions unspun.)

In three dimensions, the main rotor projects onto the plane as a circle. The spinning rotor works by trapping the air under the blade, and that one might have lift because air can not escape except through the circle. In four dimensions, the rotor takes the form of a cylinder. Now, it is perfectly legitimate for the cylinder to spin. But what happens at the end. There is nothing trying to keep the air from escaping the ends of the cylinder, so it might tip in the direction of the axis of the cylinder.

An air-screw could work, in the instance where there are several rotors in different axies, but the blades are kept in sync. Under this case, one could trap the end of one cylinder by the rotation against another cylinder, rather like the crind or tripple-crossing of cylinders. That might indeed work. The blade would not sweep a circle, but rather a kind of bloated rhombo-dodecahedron.
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Re: 4d Helicopters

Postby Prashantkrishnan » Sat Jan 17, 2015 6:40 am

Is the bicircular tegum an irregular duocylinder?

And also, the blades of the fan in 4D need not necessarily rotate around a plane tracing out a circle, but could roll around an axis tracing out a sphere. The realmic cross-section of this may look like this then (without the spherations):
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Re: 4d Helicopters

Postby quickfur » Sat Jan 17, 2015 4:02 pm

Prashantkrishnan wrote:Is the bicircular tegum an irregular duocylinder? [...]

The bicircular tegum can be thought of as the dual of a duocylinder, i.e., replace the two bounding torus-shaped surface patches of the duocylinder with two circles, which lie in orthogonal planes, and take the convex hull of that. It's the pyramid product of two circles, whereas the duocylinder is the cartesian product (prism product) of two circles.
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Re: 4d Helicopters

Postby Prashantkrishnan » Tue Jan 20, 2015 2:43 pm

quickfur wrote:
Prashantkrishnan wrote:Is the bicircular tegum an irregular duocylinder? [...]

The bicircular tegum can be thought of as the dual of a duocylinder, i.e., replace the two bounding torus-shaped surface patches of the duocylinder with two circles, which lie in orthogonal planes, and take the convex hull of that. It's the pyramid product of two circles, whereas the duocylinder is the cartesian product (prism product) of two circles.


I do not understand many of these terms. For example, I don't know what a convex hull or a pyramid product is. And also, what does a dual mean for shapes other than polytopes?

quickfur wrote: the bicircular tegum (Cartesian product of a circle with smaller circle)


If it is the Cartesian product, then won't it just be a duocylinder with one radius smaller than the other? Or is it the pyramid product only and not the Cartesian product?
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Re: 4d Helicopters

Postby quickfur » Tue Jan 20, 2015 3:47 pm

Prashantkrishnan wrote:
quickfur wrote:
Prashantkrishnan wrote:Is the bicircular tegum an irregular duocylinder? [...]

The bicircular tegum can be thought of as the dual of a duocylinder, i.e., replace the two bounding torus-shaped surface patches of the duocylinder with two circles, which lie in orthogonal planes, and take the convex hull of that. It's the pyramid product of two circles, whereas the duocylinder is the cartesian product (prism product) of two circles.


I do not understand many of these terms. For example, I don't know what a convex hull or a pyramid product is. And also, what does a dual mean for shapes other than polytopes?

quickfur wrote: the bicircular tegum (Cartesian product of a circle with smaller circle)


If it is the Cartesian product, then won't it just be a duocylinder with one radius smaller than the other? Or is it the pyramid product only and not the Cartesian product?

That must have been a typo. A tegum is not a cartesian product.

A convex hull is basically the smallest convex shape that contains the given set of points (or larger objects). For example, the convex hull of 3 non-colinear points (i.e., points that don't lie on a straight line) is a triangle. The convex hull of 4 non-coplanar points (i.e., points that don't lie on the same plane) is a tetrahedron. Generally speaking, the convex hull of the vertices of a convex polytope is the polytope itself. However, this is not true of non-convex polytopes. The convex hull of a 5-pointed star, for example, is a pentagon. Basically, you can think of the convex hull operation as "shrinkwrapping". It produces the smallest convex shape (the "wrapping") that contains the starting shape(s).

To construct a bicircular tegum, first construct two circles (conventionally of equal radius) and place one in the XY plane (by convention centered on the origin), and the other in the ZW plane (also centered on the origin). Now take the convex hull, i.e., shrinkwrap the two circles with a 3-manifold that produces the smallest convex shape that contains both circles. That's the bicircular tegum.

Now when we speak of duals for non-polytopic objects, we're generalizing the "traditional" concept of dual. As applied to a polytope, the dual operation takes the k-dimensional elements of an n-dimensional polytope and substitutes them with (n-k)-dimensional dual elements. For example, an octahedron has 6 vertices, 12 edges, 8 triangular faces. To make the dual, you replace the 6 vertices with 6 squares (which are dual squares to the vertex figure of the octahedron), the 12 edges with "dual" edges (i.e., edges that are perpendicular to the original edges), and the 8 triangular faces with 8 vertices.

As applied to the duocylinder, however, it's not immediately obvious how to perform the dual operation; however, one approach is to consider the duocylinder as the limit of n,n-duoprisms as n approaches infinity. Since n,n-duoprisms are polytopes, we can take their duals. So we can define the "dual" of the duocylinder as the limit of the dual of n,n-duoprisms as n approaches infinity. This limiting shape is the bicircular tegum.
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Re: 4d Helicopters

Postby ICN5D » Wed Jan 21, 2015 1:39 am

Now I understand the bi-circular tegum. That one was a fog until now. So, it's kinda like a 4D version of a crind. The 'shrinkwrapping' description is what worked, visually. I wonder what the implicit definition would be? Surely, there's a way to procedurally derive it, much like the other new shapes I did recently. Anybody look into functions for tegums, preferably non-parametric? Parametric is okay, but I don't understand how it works as much as implicit.
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Re: 4d Helicopters

Postby wendy » Wed Jan 21, 2015 5:25 am

I invented tegums.

Tegum is a cover, like thatch. You place the various things you want to make a tegum out of, in orthogonal spaces and a common centre, and cover them. The dimesion of the product is the sum of dimensions of the elements.

You take for example, a pentagon in the xy plane, and a line on the z-axis from +z to -z, and the cover of these gives a pentagonal tegum. It's usually called a bipyramid, but in 4D, this can confuse, because a bipyramid could be constructed as a pyramid on a pyramid, and in five dimensions, you could take bipyramid as pentagon-pentagon-pyramid, so the bi- bit refers to two pentagonal apexes.
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Re: 4d Helicopters

Postby Prashantkrishnan » Wed Jan 21, 2015 9:18 am

So the pyramid product means taking two shapes orthogonally and getting their convex hull to form a tegum
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Re: 4d Helicopters

Postby Klitzing » Wed Jan 21, 2015 1:24 pm

No! Pyramid product and tegum product intensionally are quite different. :!:

The most simple pyramid to grasp as 3D being is a tetrahedron, i.e. a vertex as tip atop of the triangular base. The most simple 3D tegum is the octahedron, i.e. the convex hull of an equatorial square and a perpandicular line segment. - But be aware, in the former product the elements are relatively shifted, whereas in the latter both defining elements are centered at the same point. :nod:

:arrow: Accordingly the general definition of the tegum product takes any 2 (convex) shapes of arbitrary dimiension each, places them in completely orthogonal subspaces of an embedding space having the sum of those individual dimensions, and aligns them with their "center" at the (highdimensional) origin. And finally it applies a convex hull operation apon that result.

:arrow: The general definition of the pyramid product rather asks for an embedding space of the sum of the individual dimensions plus an extra shift dimension. The remainder then is again quite similar: Place the (convex) "bottom element" again "centered" within one subspace, and place the (convex) "top element", also "centered", within a completely orthogonal subspace. But now take these subspaces each and pull those with respect to that extra dimension some distance apart. (That respective distance either might be provided by some given height itself, or e.g. by the size of the finally to be produced lacing edges.) And finally it also applies a convex hull operation apon that result.

(In either definition convexity of defining elements is asked only because the final usage of convex hull operation. Both definitions surely can be expanded to non-convex defining shapes, provided it can be made clear how that convex hull operation has to be generalized within the given case.)

That is
  • dim(P x_tegum Q) = dim(P) + dim(Q)
  • vertex_count(P x_tegum Q) = vertex_count(P)a) + vertex_count(Q)b)
  • edge_count(P x_tegum Q) = edge_count(P)c) + edge_count(Q)d) + vertex_count(P)*vertex_count(Q)a)b)
  • etc.
subject to:
a) dim(P) > 0, b) dim(Q) > 0, c) dim(P) > 1, d) dim(Q) > 1, ...

whereas
  • dim(P x_pyramid Q) = dim(P) + dim(Q) + 1
  • vertex_count(P x_pyramid Q) = vertex_count(P) + vertex_count(Q)
  • edge_count(P x_pyramid Q) = edge_count(P) + edge_count(Q) + vertex_count(P)*vertex_count(Q)
  • etc.

But finally the defining elements themselves become pseudo elements of the result for the tegum product (i.e. become internal faceting elements only - this is the deeper reason for the above mentioned subjections), whereas they remain true bounding elements for the pyramid product: E.g. the square only is equatorial within the octahedron, but the triangular base remains a true face of the tetrahedron.

The neutral element for the tegum product then is the point. But for the pyramid product it would be the empty set (nulloid with dimensionality -1).


One further example, easy to grasp:
The tegum product of 2 line segments would be a (2D) square (or more general: a rhomb). - Whereas the pyramid product of 2 line segments would be a (3D) tetrahedron (in its orientation as digonal antiprism: line segment atop orthogonal line segment).

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Re: 4d Helicopters

Postby quickfur » Wed Jan 21, 2015 4:12 pm

Ah, sorry, I wrongly conflated pyramid product with tegum product... what I had in mind was tegum product, not pyramid product. Mea culpa.
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Re: 4d Helicopters

Postby quickfur » Wed Jan 21, 2015 4:16 pm

ICN5D wrote:Now I understand the bi-circular tegum. That one was a fog until now. So, it's kinda like a 4D version of a crind. The 'shrinkwrapping' description is what worked, visually. I wonder what the implicit definition would be? Surely, there's a way to procedurally derive it, much like the other new shapes I did recently. Anybody look into functions for tegums, preferably non-parametric? Parametric is okay, but I don't understand how it works as much as implicit.

I'm not 100% sure, but a first guess would be something like: √(w²+x²) + √(y²+z²) ≤ C. The summation in the middle is analogous to the implicit definition of the octahedron as x+y+z ≤ C (or the 16-cell as w+x+y+z ≤ C). I'm not 100% sure it produces the correct shape in this case, but my guess is that it should.
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