# Surface Constructions

Each of the surfaces below is created by adjoining the edges of a square pattern (sometimes with a twist, sometimes without one). In the "Pattern" diagrams, we indicate this by coloring edges the same color if they need to meet each other (that is, gray meets gray, black meets black). Twists (or the lack thereof) are indicated by arrows on the edges - to correctly assemble the pattern, when the edges have been joined, arrows with the same color should point the same way.

### Construction of a Sphere

The sphere is the first of two orientable surfaces we will examine. While there are infinitely many representations of the sphere (as there are for any topological surface), we will use the most familiar one.

If you look at the square pattern for the sphere, you might notice that the left and top sides mirror the right and bottom sides. Because of this, we may actually think of this pattern as having only two sides (where the left-top is on side, and the right-bottom is the other). In this manner, we can then see that the sphere is constructed by simply joining each of the two sides of the pattern together without any twists. In the animation, we expand the surface downward, and then just align the two edges to meet each other.

### Construction of a Torus

The torus or doughnut is our other orientable surface. In its standard representation, it looks just like one would expect a doughnut to look.

To build a torus from a square pattern, we join opposite edges in pairs without twisting. First, we join two opposing edges, bringing the surface up together into a cylinder. Then we simply bend the ends of this cylinder around and join them.

To build a torus from a disk with a hole (an edge), we simply expand the surface downard between the edges, and stretch the two edges to meet each other.

### Construction of a Figure Eight Torus

The figure eight torus is an immersion of the torus which has a single (circular) double curve down the middle. Though rather uncommon (as the torus has an embedding in three space with no self intersections) it is useful to demonstrate that even a "simple" surface may have more complex (but equivalent) immersions.

To form the figure eight torus, we first form a figure eight cylinder by bringing the back edge up and forward and the front edge down and back, joining them both in the middle (and passing the surface through itself). We then simply join the two ends of this cylinder together.

### Construction of a Moebius Strip

The moebius strip is probably our simplest non-orientable surface. It is also - because of its construction - referred to as the "1-twist strip".

The moebius strip is formed from a strip (or square - remember, the two are equivalent topologically) by joining two opposing edges with a single twist. In the first animation, we first twist the strip, and then wrap the strip around to bring the edges together. The second animation combines these two steps into a single process.

### Construction of a Cross Cap

The cross cap is very closely related to the moebius strip (be sure to see "Moebius Strip to Cross Cap with a Disk Removed" in Transformations). It is one of the many representations of the projective plane, and another one of our non-orientable surfaces.

The Cross Cap pattern joins antipodal points. That is, if we were to draw straight lines through the center of the square, the endpoints of each line need to be joined.

There are actually a number of ways to join these points. In both of our constructions, we first stretch the sheet downward to form a bowl-like surface.

In the standard construction, we then bring two antipodal points to meet each other in the center. We continue to join antipodal points (stretching the surface upward) until half of the edge has been used. We then join the remainder of the edge by passing it through the piece of the surface we had just joined.

In the alternate construction, we perform what we have called a Type V move, that is, passing the surface through itself along one side. This forms a vertical pinch at the middle of the surface, leaving the formerly circular edge in the form of a figure eight. We then simply stretch the halves of the figure eight upward to meet each other.

### Construction of a Klein Bottle

The Klein Bottle (originally constructed by Felix Klein) is probably one of the best known non-orientable surfaces.

The pattern for a Klein Bottle requires that we join two of the sides with a twist, and the other two without one. To do this, we first join the twistless sides to make a cylinder. Since the ends of this cylinder must now be joined with a twist, we cannot simply connect them as we would for a torus. Instead, we must taper one end of the cylinder and pass this end through the cylinder first. Then when we join the "inside" and "outside" cylinder ends, we join antipodal points as we desired.

### Construction of a Symmetric Klein Bottle

As the standard representation of the klein bottle is rather asymmetric (and somewhat unwieldy), we sometimes find it useful to use this more symmetric version.

We form a cylinder just as we did for the normal klein bottle, and then bend around as if to form a torus. However, instead of simply bending the edges around to meet, we push the two ends through each other so that they meet along the same circle. Smoothing this surface slightly, we can see that the surface now intersects itself along a curve that resembles a parabola.

### Construction of a Pinched Torus Klein Bottle

The pinched torus klein bottle is also more symmetric than the standard representation, and, as it turns out, seems to be quite useful in some of the transformations (be sure to see Klein Bottle to Cross Caps in Transformations).

This klein bottle is formed in a manner similar to the symmetric klein bottle, except that once we have bent the two ends around, we join them by first connecting the bottom of the left side to the top of the right. When we then join the left's top to the right's bottom, we form a single double curve and two pinch points.

(Arrows are not included in this animation because matching the arrows actually requires a half-twist in the cylinder before matching up the ends, and this sort of twist makes the animation difficult to view.)

### Construction of a Figure Eight Klein Bottle

In our search for symmetry in a klein bottle, the figure eight version is probably ideal. It forms a symmetric ring that resembles a torus made of two twisting tubes.

To form this klein bottle, we first form a figure eight cylinder in the same way as we did for the figure eight torus. We then join the two ends of this cylinder with a single half twist (in the case of this animation, a quarter turn in one direction on one side, and a quarter turn in the other direction on the other side).