Moving legs while walking

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.

Moving legs while walking

Postby anderscolingustafson » Sat Mar 15, 2014 8:02 pm

In 3d land animals have their symmetry along a line and when moving they can move their legs in a none random way if they move the right leg on one side and their left on the other. In 4d land animals would tend to have their symmetry along a plane and their sides of symmetry could be arranged along a circle of symmetry. In 4d a tetronian could move it's legs in a none random way by having a wave of motion that goes along the circle of symmetry so that it moves one leg then, the leg next to it, then the leg next to the second leg, and so on until the wave of motion comes back to the first leg. In 5d the problem for a pentonian is that it would have spherical symmetry. In 5d if there was a wave of motion going from one side of symmetry to another it wouldn't reach all the sides of symmetry as some sides of symmetry would be outside the circle the wave of motion would be going in. How would a pentonian move it's legs and how might the sides of symmetry be oriented for a pentonian?
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Re: Moving legs while walking

Postby quickfur » Sat Mar 15, 2014 10:27 pm

Walking gaits in the various dimensions are an interesting topic.

In 3D, four-footed animals have a few different gaits: (1) galloping, where the front legs move together and the back legs move together, (2) symmetric trotting, where the front left leg pairs up with the rear right leg, resp. front right leg with the rear left leg, for a kind of alternating pattern, and (3) asymmetric trotting, where legs on one side of the animal move together in opposition to legs on the other side. Dogs, for example, exhibit both kinds of gaits: trotting for low to medium speed movement, and bounding for fast movement. Of course, for humans the legs have to move separately in both walking and running, otherwise motion is slow and difficult to balance (e.g., in a sack race where the legs are artificially bound together).

In 4D, it has been suggested that two legs are not enough to easily maintain balance, so in all likelihood even upright-standing creatures would need at least 3 legs. According to this logic, in n dimensions one would need at least (n-1) legs to be able to stand reliably. It's not clear, however, how balance would be best maintained with an odd number of legs; one better way I've thought of is to have 4 legs, in a kind of gait that's a cross between 2-legged walking and 4-legged trotting: the 4 feet would be placed on the vertices of a tetrahedron, which gives maximum stability in 4D, oriented such that the direction of motion is perpendicular to the bilateral symmetry of the tetrahedron (i.e., as a digonal antiprism rather than a triangular pyramid). To move forward, one would lift up the pair of legs that lie on the back edge of the tetrahedron, move them forward past the other two legs, and put them back on the ground ahead, such that the 4 legs now form a new tetrahedron, which is original tetrahedron reflected across the front edge. This is a single step. Then for the second step, the original front edge (which is now the back edge) is moved forward past the other pair of legs, thus forming another tetrahedron farther forward. So the footsteps would be an alternating pattern of pairs of alternating orientation -- in CD notation, it would be x2o -- o2x -- x2o -- o2x -- .. etc. Or, if you like, the alternation of a chain of cubes stacked by their opposite faces. This resembles 4-legged trotting, in that the legs move in pairs, but also resembles 2-legged walking, because when one pair of feet are lifted up, the other pair spans only the space of a line, so it requires the same kind of balancing that is needed in 3D when humans walk (e.g., when one leg is lifted up and the other is touching the ground).

This gait allows maximum stability when the creature is standing still, and an easy transition to/from the walking stances. It also allows changes in direction, by pairing up the legs differently, which would allow motion along a different axis of tetrahedral symmetry. In fact, the possible directions are directed along the 3D axes of the groundspace, so the creature does not have to turn to walk around along any of these grid directions! In practice, though, I expect that it's more likely to reserve a single pairing for forward/backward motion, and the other two are retained only for sideways "strafing" style movement -- since the ability of the legs to arbitrarily pair up doesn't mean that the creature doesn't have to turn its head to look where it's going!

Of course, I'm sure there are many other possibilities in 4D and beyond. Perhaps somebody else has a better idea. :P
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Re: Moving legs while walking

Postby Polyhedron Dude » Sun Mar 16, 2014 9:07 am

Here are some leg patterns and possible walk patterns for 3 to 5 dimensional "humans".

Three-Dimensional

Dyadic leg pattern - that's us, 2 legs at the vertices of a dyad, we put one foot in front of the other - foot A then foot B, foot A, B, etc.

Four-Dimensional

Triangular leg pattern - this tetronian has trigonal symmetry and therefore three legs, his walk pattern will likely be in this pattern: A,B,C,A,B,C,A,B,C,...

Square leg pattern - this tetronian has either rectangular or square symmetry. He could have the following walk patterns: cyclic 4, bowtie, rectangular, or rhombic. Let A,B,C,D represent the four legs, where AB, BC, CD, and DA forms the lines of a square. Cyclic 4 would have this pattern A, B, C, D, A, B, C, D. Bowtie would go like A, B, D, C, A, B, D, C. Rectangular would move two legs at a time: AB, CD, AB, CD. Rhombic would move opposite legs at the same time: AC, BD, AC, BD. Rectangular and rhombic are the most likely walking methods for a square tetronian.

Pentagonal leg Pattern - This tetronian has pentagonal symmetry and has 5 legs. Possible walking patterns could be cyclic 5, cyclic 5/2, and "three-two". Cyclic 5 would have a A,B,C,D,E pattern and 5/2 would have a A,C,E,B,D pattern. Various "three-two" patterns would work - here's one: ABD, CE, ABD, CE.

Hexagonal leg pattern - This tetronian most likely has trigonal symmetry, but could have hexagonal. Most likely walking patterns would be the star-of-David walk and the "asterisk" walk. The star of David would go like this: ACE, BDF, ACE, BDF. The asterisk walk would go: AD, BE, CF, AD, BE, CF.

Five-Dimensional

Tetrahedral leg pattern - this pentonian has tetrahedral symmetry with four legs and would likely walk with a "disphenoid" walk - AB, CD, AB, CD

Octahedral leg pattern - this dude has either tetrahedral or octahedral symmetry with 6 legs arranged like the vertices of an octahedron. He would likely walk with a trigon antiprism pattern.

Cube leg pattern - this dude has 8 legs arranged like a cube, he may either walk in a square prism pattern or a stella octangula pattern - both moving four legs at a time.

Icosahedral leg pattern - has 12 legs. Most likely walk pattern may move four legs at a time, where these four legs form a golden rectangle going through the center of the ike.

Dodecahedral leg pattern - has 20 legs (this may be a bit overkill on number of legs needed, so may not get chosen as a 5-D human design). He would likely move 4 legs at a time, each four legs forming a tetrahedron in a 5-tet compound.
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Re: Moving legs while walking

Postby wendy » Sun Mar 16, 2014 10:39 am

Ian Stewart in 'fearful symmetry' gives a discussion on the use of oscillators to create gaits, and about the changing of gaits from, eg amble to walk to pronk to gallop. Some discussion on cockroach gates, which includes two- and three- step gates. So i imagine it is the same in four dimensions too,
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Re: Moving legs while walking

Postby Teragon » Wed Aug 17, 2016 5:14 pm

Quickfur, this is exactly how I imagined walking in 4D. I think it's the way that's the easiest one to imagine, it feels like a funny mixture of walking and standing. Orientation is simpler with four legs, because the lateral directions are equivalent and each direction has its associated limbs (left-right, ana-kata) than with three legs, where left-right- and ana-kata-symmetry is broken (stepping to the left feels different than stepping to the right). I think the alternating motion of two pairs of legs is easier to imagine, but also faster to navigate than three seperate legs. It's more similar to 3D in anatomy and movement, so that it's easy to learn, yet different (raising legs on opposite sides at once) so that relapsing into 3D thinking is not a big obstacle.

Walking straight has been explained already. Typically there are more ways to achieve the same result than in 3D, but I'm gonna concentrate on what seems to be the simplest and fastest way to do it:

Walking an arc
To walking an a arc, say, to the right, each time you do the step with the left und the right leg, you just have to step a little bit further with the left leg than with the right leg. The ana-kata step just follows straight, slightly adjusting the orientation in the 2-plane of the arc. Arcs in arbitrary directions are traced out by legs of both pairs stepping different lengths, where each pair of legs contributes the component of the targeted direction in the direction of the pair to the turn. Another way to do it would be to first turn in the lateral 2-plane until the direction you want to turn to is aligned with one pair of your legs and then just use this pair.

Turning in the lateral 2-plane
It's a more or less redundant possibility in 4D. You raise your left and right leg and shift one slightly to ana, one slightly to kata. Bringing the body again to a neutral position and a bit further, the other two legs follow, one stepping now a bit more to the left, one to the right. And so on.

Sidestepping
To step to the right, the most natural possibility might be, taking a broad stand, to step to the right with the ana and kata leg so that they form a line with the right leg, shift the weight onto those legs and take a step to the right with the left and the right leg until the left leg forms a line with the legs of the other pair.

Straddling the legs
Shift you weight towards one leg of each pair and sidestep with the other two legs simultaniously. Alternatively: Jump and move all four legs apart.
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Re: Moving legs while walking

Postby gonegahgah » Thu Aug 18, 2016 7:13 am

Certainly a pet subject of mine Teragon.
Creatures come up with interesting combinations of movements based upon the number and movement of legs that they have.
Multi-legged animals in our world certainly find different ways to walk that are unavailable to us.

We have to settle for our bi-legged running that involves shifting from one leg on one side to the other.
Kind of an oscillating motion and distribution of weight though we do well to keep it fairly linear.
We of course additionally have things like skipping and hopping (and a weird thing called a triple jump).

My personal favourite for 4D is for the top intelligence to have 3 legs as an expectation of evolutionary minimalism.
Motion then is like ours but cyclic with each leg handling its moment against the minimal three opposite points of falling.
It still oscillates like us but with a circular motion rather than a bi-lateral motion. Might even to a degree be less rigorous on their bodies than our motion.
It does reduce the possible motions but they could still have jake-the-peg motions, double hop, etc.
They could also run from leg A to B to C to A or, from leg A to C to B to A.
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