Notices by Danpiker (danpiker@mathstodon.xyz)

and here's a version clipped with a moving plane to reveal more of the interior spiralling structure

https://skfb.ly/puHHF

The surface shown in my last few posts can be described as a diagonal on the hypersphere. In this 🧵 I'll expand a bit on what I mean by that.

On a sphere we can trace 'loxodrome' curves of constant bearing, diagonal to the longitude/latitude directions, spiralling from one pole of the sphere to the antipodal one.
https://en.wikipedia.org/wiki/Rhumb_line

Stereographically projecting from the top pole onto a horizontal plane gives logarithmic spirals, where one of the poles projects to the point at infinity. However, if instead we turn the sphere on its side before projecting we get a family of beautiful double spiral curves which reveal the true symmetrical nature of the 2 poles.

What is the equivalent of these curves in one dimension up?

Instead of the familiar sphere in 3d space a.k.a. the 2-sphere (which can be parameterised with 2 coordinates, longitude and latitude) we now have the 3-sphere or hypersphere in 4d space.

One particularly nice way of parameterising the 3-sphere is using Hopf coordinates (https://en.wikipedia.org/wiki/3-sphere#Hopf_coordinates).

Stereographic projection which maps the 2-sphere to the flat plane can also be generalised to higher dimensions, and it conformally maps the 3-sphere to our familiar Euclidean 3d space.

This projects Hopf coordinates to a triply orthogonal system composed of spheres, planes and nested tori. At one extreme the torus becomes a horizontal circle and at the other a vertical line (or circle through infinity) through its centre (see toroidal coordinates here https://mathcurve.com/surfaces.gb/tripleorthogonal/tripleorthog.shtml).

However, like the lower dimensional case, we can also apply a 4-dimensional rotation to the 3-sphere before projecting, so that this circle and straight line become a symmetrical pair of linked circles. My earlier posts show the resulting triply orthogonal system.

https://mathstodon.xyz/@Danpiker/113625825121929332

https://mathstodon.xyz/@Danpiker/113636827127103066

To get an equivalent in 3d to the projected loxodrome *curve* in 2d, we now want to take a diagonal *surface* in this coordinate system.
In 3 dimensions this could mean either a diagonal of 2 directions or a diagonal of all 3 directions.

Taking the diagonal of only 2 directions gives the surface shown here:

https://mathstodon.xyz/@Danpiker/111727832885683368

https://skfb.ly/oPFYM

Notice how this wraps around one circle like a smoke ring, while at the other circle it becomes a tapering tube nesting into itself like an Ouroboros.

(When one of the circles is a straight line, this can also be seen as the continuation of the surface in Escher's 'Spirals' engraving if it kept wrapping around itself until it filled all of space).
However, this is not really a surface equivalent of our original projected loxodrome curve, because it lacks that symmetry of the 2 poles, instead meeting the 2 circles in different ways.

Only when we take the diagonal of all 3 directions do we get the symmetry we want between the circles, meeting each of them in the same way that both tapers and twists, similar to the way Dini's surface (https://mathcurve.com/surfaces.gb/dini/dini.shtml) meets a straight line.
Note that for visibility I am trimming the surface to show a finite region in my animations, but it actually spirals through all of space, with the only singularities being this symmetrical pair of linked circles. Also, like with the loxodrome, we can vary the angle to change how many turns it takes as it goes from one circle to the other, more turns producing a denser filling of space.

The first post in this thread links to a rotatable 3d model of the surface. It's quite hard to grasp the shape from just an image or even an animation, I encourage you to turn it around yourself and try following a path from one circle to the other.

Here's another version of the model with a slightly tighter wrapping angle:
https://skfb.ly/puJDz

Gyroid 7-direction pencil holder. Thanks to @standupmaths for the pencils!

The square pencils fit in the existing orthogonal channels through the gyroid, while the hexagonal ones have holes cut for them and pass through the vertices of the 2 laves graphs which are the skeletons of the surface (as shown here https://vimeo.com/2553639)

Fully assembled! It took a bit of force to get the hexagonal ones in. If I was printing it again I think I'd scale it up by a few %