My 4D world's equivalent of Tiktaalik, the ancestor of all terrestrial endoskeletal animals, is a tetralaterally symmetric octopod with two limb girdles holding four limbs in the front and four limbs in the back. (Note that because this is in 4 dimensions, the planes of radial symmetry in which the limbs are attached are fully perpendicular to the plane formed by the spinal axis and the vertical axis.)

Among 3D animals, it seems pretty much universal for jointed limbs to have a joint with maximal freedom for the shoulder / hip (a ball joint for tetrapods, and a compound hinge joint for arthropods), then a simple hinge join that can only rotate in a single plane (i.e., an elbow or knee), and then variously complicated stuff for the terminal segment of the limb--i.e., extended tarsals/metatarsals than turn the wrist/ankle into an additional or reversed elbow/knee equivalent, all sorts of different foot/fin/wing structures, etc. But all the specialized stuff aside, you pretty much always see a maximally generic joint with the body, then one or two simple hinges.

It is easy to generalize that to 4 spatial dimensions. The only big change it that you need at least 4, rather than 3, muscle groups to control the full range of motion of the shoulder / hip joints. But is that actually the ideal structure for a 4D limb? Do we need full range of motion of the shoulder / hip in 4D, or can it be limited to a 3D range (being a hybrid between a hinge and ball joint that only exists in 4 dimensions) with only 3 controlling muscle groups? And if we do need the full geometrically possible range in the shoulder and hip, how about the elbow and knee? Can those remain simple hinges anyway, or do they need greater freedom as well?

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    $\begingroup$ An octopus trying to mimic Klein bottle... $\endgroup$ – user6760 Nov 29 '19 at 5:19
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    $\begingroup$ A carrot shaped cat? $\endgroup$ – candied_orange Nov 29 '19 at 16:09

Let us consider the needs of locomotion in 2, 3 and 4 physical dimensions of a creature with legs walking upon an n-sphere.

The simplest situation is a creature in 2 physical dimensions walking upon the edge of a solid disk. Regardless of the number of legs, the greatest efficiency of motion is achieved with 2 or more legs, each leg having a foot joint that allows the foot to align with the surface, a knee joint that allows the leg to lengthen or shorten, and a hip joint around which muscular force may produce the greatest angular movement of the whole limb for the least effort.

In 2d, all three joints need only be simple hinge joints, as there is only one degree of freedom.

In 3d, the foot must be able to pitch, roll and yaw in order to place the foot on the ground properly. The hip must also be able to flex in two dimensions as well as being able to rotate. The knee joint still only needs to flex in one plane in order to shorten the leg, though that it flexes such that it can provide assistance to forward movement is a bonus of which many but not all creatures take advantage.

So, by extension, in 4d, the foot would need to be able to rotate in order to engage with the ground and the hip would need to be able to rotate to provide motive force in all three dimensions of the ground surface. The knee would still need to flex in a single plane on order to shorten the leg, and could be oriented to provide mechanical advantage to forward movement.


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