"Math Horizon" is a view of a twodimensional sphere immersed in fourspace
so that it has exactly one point of selfintersection. To see how this
works, first note that surfaces in fourspace generally intersect in points
rather than in curves, as they do in three space. For example, if we label
the axes x, y, z, and w, then the xyplane
and the zwplane are twodimensional planes in four space, but they
intersect in only one point: the origin.
To form the sphere depicted in "Math Horizon", we began by taking the unit
disc in the xyplane and the unit disc in the wzplane; since
they intersect in a single point, these form the essential
selfintersection in the surface. The trick now is to attach the
boundaries of these two discs so as to form a sphere, and in such a way
that no additional self intersection is produced.
The boundaries are two circles, which can be parameterized as (cos q, sin q, 0, 0) and
(0, 0, cos q,
sin q). For a
given q, these two points, together with the
origin, determine a plane in fourspace (think of the points as vectors
based at the origin that span the plane). For different values of q, these planes intersect only at the origin, so if,
for each q, we connect the two boundary points
by a curve lying in this plane, we will have joined the two disc boundaries
to form a sphere with no additional selfintersection, as desired.

The two points (cos q, sin q, 0, 0) and (0, 0,
cos q,
sin q) can be
joined by circular arcs (left). A smooth figureeight can replace the
piecewise curve (right). 
Note that the two points, when considered as vectors at the origin, are
perpendicular unit vectors, so they act just like the unit x and
yaxes in the xyplane. The intersection of the plane
spanned by these vectors and one of the discs would be the segment from
1 to 1 along the xaxis, and with the other,
the corresponding segment on the yaxis. These two segments form a
"cross" at the origin, and one natural way to attach them is by two
circular arcs thus forming a figure8 with an axis of symmetry along the
line y = x. A piecewisedefined version of the
twosphere in fourspace can be produced in this way. On the other hand,
we could form a smooth version of the surface if we had a smooth (rather
than piecewisedefined) figure8.
The equation (cost, sin 2t) parameterizes a figure8 that has
the xaxis as an axis of symmetry, though the equation (cost,(1/2) sin
2t) = (cost, sint cost) = cost (1, sint) is more aesthetically pleasing, as
the lobes of the figure8 are rounder and cross at an angle of 90 degrees.
Rotating this curve by 45 degrees about the origin produces a smooth
figure8 with its axis along the line y = x and
its crossing tangent to the x and y axes, as desired. Using
a standard rotation matrix with angle f =
p/4, we obtain



 =  
 
cost 

æ è 

 
ö ø 

æ è 

 
ö ø 
 =  
 
cost 

æ è 

 
ö ø  . 

 

Writing this in vector notation, we find
(x,y) 
 =  
 
cost 

[(1  sint) (1,0)
+
(1 + sint) (0,1)].

 

Now, replacing the vectors (1,0) and (0,1) by the two vectors from the
boundary of the discs in fourspace gives a smooth parameterization by
t and q of the twosphere in fourspace
with exactly one point of transverse selfintersection:
(x,y,z,w) 
 =  
 
cost 

[ (1  sint)
(cos q,
sin q, 0, 0)
+
(1 + sint)
(0, 0, cos q,
sin q) ].

 

Note that this surface lies within the unit sphere in fourspace and
touches the unit sphere when t = 0, namely along the
curve
(x,y,z,w) 
 =  
 
(cos q,
sin q,
cos q,
sin q)

 

a circle on the foursphere. The image shown in "Math Horizons" is the
stereographic projection of this surface from the point on this circle
where q = 0. Because the surface
passes through the point of projection, it's image appears to extend out to
infinity in threespace. Bands of the surface have been removed to help
make the structure of the surface and its parameterization more apparent.