"Math Horizon" is a view of a two-dimensional sphere immersed in four-space
so that it has exactly one point of self-intersection. To see how this
works, first note that surfaces in four-space 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 *xy*-plane
and the *zw*-plane are two-dimensional 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 *xy*-plane and the unit disc in the
*wz*-plane; since they intersect in a single point, these form the
essential self-intersection 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 , sin , 0, 0) and (0, 0, cos , sin ). For a given , these two points, together with the origin, determine a plane in four-space (think of the points as vectors based at the origin that span the plane). For different values of , these planes intersect only at the origin, so if, for each , 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 self-intersection, as desired.

The two points (cos , sin , 0, 0) and (0, 0, cos , sin ) can be joined by circular arcs (left). A smooth figure-eight 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 *y*-axes in the *xy*-plane. The intersection of the plane
spanned by these vectors and one of the discs would be the segment from
-1 to 1 along the *x*-axis, and with the other, the corresponding
segment on the *y*-axis. These two segments form a "cross" at the
origin, and one natural way to attach them is by two circular arcs thus
forming a figure-8 with an axis of symmetry along the line
*y* = *x*. A piecewise-defined version of the
two-sphere in four-space 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 piecewise-defined) figure-8.

The equation (cos *t*, sin 2*t*) parameterizes a
figure-8 that has the *x*-axis as an axis of symmetry, though the
equation (cos *t*, (1/2)sin 2*t*) = (cos *t*,
sin *t* cos *t*) = cos *t* (1,
sin *t*) is more aesthetically pleasing, as the lobes of the
figure-8 are rounder and cross at an angle of 90 degrees. Rotating this
curve by 45 degrees about the origin produces a smooth figure-8 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 = /4, we obtain

Writing this in vector notation, we find

Now, replacing the vectors (1,0) and (0,1) by the two vectors from
the boundary of the discs in four-space gives a smooth parameterization
by *t* and of the two-sphere in
four-space with exactly one point of transverse self-intersection:

Note that this surface lies within the unit sphere in four-space and
touches the unit sphere when *t* = 0, namely along the curve