Taking cross-sections of a shape is a good way to understand its form and spatial extent. Shapes with nontrivial topology are particularly interesting as their slices exhibit nontrivial topologies as well. Sequencing these cross-sections through their discrete topological changes helps one understand the topology of the overall shape. This is the primary focus of Morse theory and it is especially useful when the shape being analyzed is difficult to visualize.
In this article we first look at the the torus
2

=
1
S
×
1
S
and how its topology affects the topologies of its slices. We then look at the hypertorus
3

=
1
S
×
1
S
×
1
S
=
2

×
1
S
as embedded in
4

and compute its 3-dimensional cross-sections to gain an understanding of its overall structure and topology.

Initialization code

These are some custom functions are defined which produce the graphic elements used throughout this article.

makeTorusCutGraphics[…]


makeTorusCriticalNeigrhoodGraphics[…]


makeTorusSliceGraphics[…]


makeHypertorusGraphics[…]


Torus

We start with the torus which is a 2-manifold defined as the product of two circles
2

=
1
S
×
1
S
. We parametrize the torus (with radii
r
1
<
r
2
) as a function of two parameters (θ,ϕ) mapping into
3

:
In[]:=
Block[{r1,r2,θ,ϕ},Row[{MatrixForm[{θ,ϕ}],"↦",MatrixForm[ExpandAll@{r1Cos[θ],(r2+r1Sin[θ])Cos[ϕ],(r2+r1Sin[θ])Sin[ϕ]}]}]]
Out[]=
θ
ϕ
↦
r1Cos[θ]
r2Cos[ϕ]+r1Cos[ϕ]Sin[θ]
r2Sin[ϕ]+r1Sin[θ]Sin[ϕ]
With this parametrization the torus is constructed as an extrusion of a smaller circle in the xy-plane (red) extruded along a larger circle (blue) oriented in the yz-plane:
In[]:=
Module[{torus},​​torus[r1_,r2_][{θ_,ϕ_}]:={r1Cos[θ],(r2+r1Sin[θ])Cos[ϕ],(r2+r1Sin[θ])Sin[ϕ]};​​Graphics3D[{​​ParametricPlot3D[torus[2,5][{θ,ϕ}],{θ,0,2π},{ϕ,0,2π},PlotStyle->Opacity[0.2,White],MeshStyle->Opacity[.2]][[1]],​​ParametricPlot3D[torus[2,5][{θ,0}],{θ,0,2π},PlotStyle->{Thickness[.02],Red}][[1]],​​{Blue,AbsoluteThickness[3],Arrowheads[.1],Arrow@Tube[Table[{0,5Cos[θ],5Sin[θ]},{θ,0,1.95Pi,.1}]]}​​}​​,ViewPoint->{2,0,.4},Boxed->False​​]​​]
Out[]=
Let’s start taking slices. We’ll use the z-coordinate as our height function which means we will be taking horizontal slices along planes that are parallel to the xy-coordinate plane. We start with a slice towards the bottom. The slice itself is a single curve (red) approximating an ellipse. The portion of the torus located below the slice plane (gray) has the shape of a bowl opening upwards:
Out[]=
We move the slice plane upwards we see the red curve is now pinched into a figure 8 crossing itself at the center. This corresponds to a critical point in the height function:
Out[]=
Moving up past that critical point to the equator, we see the slice has split into two separate circles. and the gray portion below the slice now has the topology of a cylinder:
Out[]=
There is another critical point where the disjoint circles come back together creating a pinch point at where they merge:
Out[]=
Near to the top the slice curve is once again a single ellipsoid and the gray portion has the topology of a punctured torus:
Out[]=
Finally, after the slice curve closes up at the top critical point we recover the whole
In total there are four critical points for this embedding and height function. The top and bottom critical points represent places where the slice curve degenerates to a point while the middle two critical points are places where the slice curve gets pinches and splits (or merges). The top and bottom critical points are have neighborhoods exhibiting positive curvature and the two middle critical points are saddle points with neighborhoods exhibiting negative curvature:
Looking just at the sequence of horizontal slices we see a single curve growing from a critical point at the bottom, then it pinches into two circles which separate and then pinch again to merge back together to a single curve, then that curve finally degenerates and shrinks to the top critical point. We understand that the story of these slices describes the topology of the torus.

Hypertorus

This parametrization starts with a small torus (red) oriented in the xyz-plane and extrudes it along a large circle (blue) in the xw-plane:
Already there is a lot going on. Notice that these slices have holes in them. This begins to make sense when one considers that the hypertorus has a void inside it that is swept out by the “donut hole” in the torus that is being extruded and revolved. This void can be plotted as a torus (shown in transparent gray) that is swept out by the inner circle of the revolved torus:
We can see that the holes of the slices follow closely to this void. Indeed this region is very much where the hypershape’s material isn’t:
Similarly, there is a larger toroidal region which shows the shape’s outer boundary:
Putting everything together we can get a sense for how the material of the shape is confined to a region between the inner void and the outer shell:
Let’s start looking at the slices and the critical points where its topology changes. We start at the bottom critical point where the slice has the topology of a sphere:
The next critical point comes at the bottom of the void region. Here the slice gets pinched at its center and becomes a genus-1 surface:
Continuing upwards the slice surface expands laterally and starts getting narrow towards the middle:
The second critical point is located at the first saddle point of the inner torus. Here the slice gets pinched again at the center:
Directly above this critical point the slice becomes a genus-2 surface:
The next critical point is the bottom saddle point of the outer torus. Here the slice is pinched again:
Just past this critical point the topology of the slice changes as the slice becomes the disjoint union of two separate genus-1 surfaces:
Continuing upwards the separate toroidal components of the slice move apart then begin moving back together after they cross the equator:
The next critical point is the top saddle point of the outer shell. The two components of the slice pinch back together:
Once again the slice becomes a genus-2 surace:
Next, at the upper saddle point of the inner torus, the slice pinches again and the two wholes of the genus-2 surface merge together:
At this level, the slice is once again a genus-1 surface
The next critical point is at the top of the inner torus and it is where the genus-1 slice pinches at the center and its hole closes up:
Finally, the slice is now a genus-0 spheroid which shrinks until it closes up around the topmost critical point:
Now we can stack multiple slices to get an overall view of the whole hypertorus. For this plot the projection is adjusted and the w-axis is tilted further into the z-direction and some (and tilted a bit towards the negative y-direction). This allows more slices to be viewed and plotted together allowing us to view the complete story of how topology of the slices changes relative to the critical points:
In this final plot we show everything together. With the outer torus showing the containing region and the inner torus showing the void, we can get a sense for how the hypertorus is shaped:

CITE THIS NOTEBOOK

Hypertorus cross-sections: Morse theory in 4D​
by Dugan Hammock​
Wolfram Community, STAFF PICKS, August 11, 2023
​https://community.wolfram.com/groups/-/m/t/2987965