The Möbius Strip, One Side and One Edge, math that surprises you lesson

Tape the two ends of a paper strip together and you get a band with an inside, an outside and two separate edges. Give one end half a turn before taping, and three surprising things happen. A pen line drawn down the middle reaches both faces of the paper without ever crossing an edge, the two edges become a single loop, and a cut all the way along the middle leaves one piece instead of two. This page explains each surprise by following which points the join connects, and shows what changes with more twists or with a cut away from the middle.

Key terms

Möbius strip

Definition Start with a rectangle of length $L$ and width $2w$ , and describe each of its points by coordinates $(x, y)$ with $0 \le x \le L$ along the length and $-w \le y \le w$ across the width, so that the center line is $y = 0$ . The Möbius strip is the surface made by joining the right end of the rectangle to its left end with the heights reversed. For every $y$ from $-w$ to $w$ , the two points

(L, y) \;\longleftrightarrow\; (0, -y)

become a single point of the surface. With paper this is done by giving one end a half-twist before taping, so that the top corner of each end meets the bottom corner of the other. Joining every $(L, y)$ to $(0, y)$ instead, with no twist, gives an ordinary band. The mathematical strip is an ideal surface with no thickness, and only the way its points are joined matters, not the way paper bends. Three facts make it surprising. The ordinary band has two sides and two edges, but the Möbius strip has only one side, and its whole edge is one closed loop. A cut along the entire center line of the ordinary band leaves two bands, but the same cut leaves the Möbius strip in one piece. That piece is a band twice as long and half as wide, with two sides and two edges, so it is not another Möbius strip. All three facts come from the reversed join, which sends a path that reaches the right end above the center line into the left end below it. The strip is named after August Ferdinand Möbius, and it was found independently by Möbius and by Johann Benedict Listing in 1858.

In plain words The half-twist plugs the top edge of the strip into its bottom edge. Travel once around the strip at a fixed distance from the center line and you come back at the same distance on the other side of it. That one swap is behind the single side, the single edge and the surprising cut.

Example Take a strip of paper 30 centimeters long and 4 centimeters wide, so $L = 30$ and $w = 2$ . The center line splits it into two lanes, the upper lane $0 \le y \le 2$ and the lower lane $-2 \le y \le 0$ . In an ordinary band each lane is joined to itself, so a cut along the center line leaves two bands, each 30 centimeters around and 2 centimeters wide. In the Möbius strip the join sends height $y$ to height $-y$ , so the right end of the upper lane meets the left end of the lower lane, and the right end of the lower lane meets the left end of the upper lane. After the cut the two lanes therefore form one band, the upper lane followed by the lower lane, measuring in centimeters

\begin{aligned} \text{length} &= 30 + 30 = 60, \\[0.4em] \text{width} &= 4 \div 2 = 2. \end{aligned}

The area is unchanged, since $60 \cdot 2 = 30 \cdot 4 = 120$ square centimeters. To see that the long band is two-sided, give each of its points the distance $u$ from the cut, so $u = y$ in the upper lane and $u = -y$ in the lower lane. Each join pairs a point at height $y$ with the point at height $-y$ in the other lane, and these two points have the same $u$ . The long band therefore closes up with no reversal, exactly like an ordinary band, and it has two sides and two edges.

Topology

Definition Topology is the branch of mathematics that studies the properties of shapes that depend only on which points are close to which, and not on lengths, angles or areas. Two shapes are topologically equivalent when there is a correspondence between their points that pairs each point of either shape with exactly one point of the other and is continuous in both directions, so that points close together in one shape correspond to points close together in the other. Bending, stretching and shrinking without tearing or gluing give an equivalent shape, and so does cutting a shape and then gluing the same points back together. A property shared by all shapes equivalent to a given one is a topological invariant. The number of pieces, the number of boundary components and orientability are invariants of surfaces. Lengths and areas are not, and neither is the number of half-twists in a band, apart from whether that number is odd or even.

In plain words Imagine the surface made of perfect rubber with no thickness. Topology keeps track of what survives any amount of bending and stretching, such as how many pieces there are, how many edges and whether the surface is one-sided, and it ignores sizes. The properties on this page belong to the surface itself, not to the way it happens to sit in space.

Example The ordinary band and the Möbius strip are not equivalent, because the band has two boundary components and the strip has one. So no amount of bending or stretching makes one of them into the other without tearing or gluing. A band joined after a full twist, on the other hand, is equivalent to the ordinary band. Cut it across, untwist it and tape the same points back together, and an ordinary band appears. The long band left by the center cut of the Möbius strip is equivalent to an ordinary band as well, since its ends are joined with no reversal. It is twice as long, half as wide and twisted in space, and topology ignores all three differences.

Half-twist

Definition A half-twist rotates one end of a strip by half a turn, 180 degrees, about the strip’s center line, before the two ends are joined. It moves the top corner of that end to the bottom and the bottom corner to the top. A full twist is two half-twists, a rotation by 360 degrees, which brings every corner back to where it started. After $n$ half-twists, in either direction or in a mixture of both, the ends are joined by

(L, y) \;\longleftrightarrow\; \bigl(0, (-1)^n y\bigr),

which is the join of an ordinary band when $n$ is even and the join of the Möbius strip when $n$ is odd. So a band with an even number of half-twists is topologically an ordinary band, with two sides and two edges, and a band with an odd number is topologically a Möbius strip, with one side and one edge. Beyond whether it is odd or even, the number of half-twists changes only the way the band sits in space.

In plain words Count the half-turns. An odd count swaps the corners and gives a Möbius strip, and an even count leaves them in place, however twisted the band looks. A band with a full twist is not a doubled Möbius strip. As a surface it is an ordinary band.

Example Label the corners of the left end $A$ at the top and $B$ at the bottom, and those of the right end $C$ at the top and $D$ at the bottom. Each half-turn of the right end swaps the places of $C$ and $D$ , which decides which corners meet.

Half-twists	Corner $C$ meets	Corner $D$ meets	Sides	Edges
$0$	$A$	$B$	$2$	$2$
$1$	$B$	$A$	$1$	$1$
$2$	$A$	$B$	$2$	$2$

Every further half-turn swaps the corners again, so the pattern keeps alternating. A half-twist to the left and a half-twist to the right give two strips that are mirror images of each other in space, and both are Möbius strips.

Non-orientable surface

Definition A small circular arrow drawn around a point of a surface picks one of the two ways of turning around that point. A surface is orientable if such an arrow can be chosen at every point so that the arrows at nearby points always turn the same way, and it is non-orientable if no such choice is possible. Equivalently, a surface is non-orientable exactly when it contains a closed path along which a small circular arrow, slid continuously within the surface, returns to its starting point turning the other way. For a surface that sits in ordinary space without passing through itself, non-orientable means the same as one-sided. A short arrow standing straight out of the surface can then be slid along such a path, always perpendicular to the surface, and it returns pointing out of the opposite face.

In plain words On a non-orientable surface there is no way to agree everywhere on which way is clockwise, and when the surface sits in space there is no way to agree everywhere on which face is the front. Going around the right loop swaps them.

Example On the Möbius strip, draw a small clockwise arrow at a point of the center line in the rectangle and slide it along the center line to the right end. Crossing the join changes height $y$ into $-y$ and keeps the direction along the strip, which is a mirror reflection, so the arrow continues from the left end turning counterclockwise. One more trip around makes it clockwise again. In an ordinary band the join keeps both directions, so clockwise arrows drawn everywhere in the rectangle match across the join, and the band is orientable. A pen line on the paper strip 30 centimeters long shows the same thing. Drawn down the middle, it comes back to its starting point after 30 centimeters but on the opposite face of the paper, and it closes up only after 60 centimeters, having run along both faces. The ideal surface has no thickness, so there the center line is a single loop 30 centimeters long, and what comes back reversed is the arrow standing out of the surface.

Boundary component

Definition A point of a surface lies on its boundary, or edge, when the small patches of the surface around it are shaped like a half-disk with the point on the straight side, rather than like a whole disk with the point in the middle. A boundary component is a connected piece of the boundary, a largest set of boundary points in which any two can be joined by a path that never leaves the boundary. For a surface glued together from finitely many polygons, such as every band on this page, each boundary component is a closed loop, and the number of boundary components is a topological invariant.

In plain words A boundary component is one edge you can run a finger along until it comes back to where it started. Counting them takes a trip all the way around, because a short section of any of these bands shows an edge on each side.

Example In the rectangle the top edge runs from $(0, w)$ to $(L, w)$ and the bottom edge from $(0, -w)$ to $(L, -w)$ . The short ends add nothing to the boundary of a band, because after the join each of their points other than the corners has a whole disk of surface around it. In the ordinary band the point $(L, w)$ is joined to $(0, w)$ , so the top edge closes on itself, the bottom edge does too, and there are two boundary components of length $L$ each. In the Möbius strip the point $(L, w)$ is joined to $(0, -w)$ , so the top edge runs on into the bottom edge, and the end $(L, -w)$ of the bottom edge is joined to the start $(0, w)$ of the top edge. The whole boundary is one loop of length $2L$ , which is 60 centimeters for the 30 centimeter strip. The long band left by the center cut has two boundary components of 60 centimeters each. One is the edge of the original strip, and the other is a new loop formed by the two banks of the cut.

Check yourself

Make a band from the same strip, 30 centimeters by 4 centimeters, with a full twist, that is two half-twists, instead of one. How many sides and edges does it have, and what does a cut all the way along its center line leave?

Show the answer

After two half-turns each corner is back in its place, so $C$ meets $A$ and $D$ meets $B$ , and the join is $(L, y) \longleftrightarrow (0, y)$ , the join of an ordinary band. The band has two sides and two edges, each 30 centimeters long, even though it looks twisted.

Each lane is joined to itself, as in an ordinary band, so the center cut leaves two separate bands, each 30 centimeters around and 2 centimeters wide, with two sides and two edges. In space the two bands are linked like two links of a chain and cannot be pulled apart without cutting, because the full twist winds the two lanes around each other once. Twisting alone does not make a band one-sided. What matters is whether the number of half-twists is odd.

Extend the pattern. Give the strip three half-twists before taping. How many sides and edges does the band have, and what does the center cut leave?

Show the answer

Three is odd, so after three half-turns $C$ is at the bottom and meets $B$ , while $D$ meets $A$ . The join is $(L, y) \longleftrightarrow (0, -y)$ , the same as for the Möbius strip, so the band has one side and one edge, a loop 60 centimeters long.

At the join the lanes swap exactly as in the Möbius strip, so the center cut leaves one piece, 60 centimeters long and 2 centimeters wide. The distance from the cut is matched at both joins as before, so this piece has two sides and two edges. The difference shows in space. Along its length the piece runs twice around the band, and the twisting, three half-turns on each trip, carries it three full times around the original center line. Its own middle line stays at a fixed distance from that center line, so it lies on a thin tube around it, and a loop on such a tube that goes twice around lengthwise while winding three times around the tube is a trefoil knot. So the long piece is tied in a knot that cannot be undone without cutting.

Change the rule for the cut. On a Möbius strip 30 centimeters long and 3 centimeters wide, start a cut 1 centimeter from the edge and keep it 1 centimeter from the edge all the way. When does the cut close, and what pieces are left?

Show the answer

Here $w = 1.5$ , so the cut starts along the height $y = 0.5$ , which is 1 centimeter below the top edge. At the join the height $0.5$ becomes $-0.5$ , so the cut continues along $y = -0.5$ , which is 1 centimeter above the bottom edge. Only after a second trip does it return to $y = 0.5$ , so it closes after 60 centimeters. The center line is the only line of constant height that closes after one trip, because $y = 0$ is the only height with $y = -y$ .

The two heights split the strip into three lanes. The middle lane $-0.5 \le y \le 0.5$ is joined to itself with the heights reversed, so it is a Möbius strip 30 centimeters long and 1 centimeter wide. At the join the top lane $0.5 \le y \le 1.5$ meets the bottom lane $-1.5 \le y \le -0.5$ and the other way around, so these two form one band 60 centimeters long and 1 centimeter wide. Measured from the cut, a point at height $y$ in the top lane has distance $y - 0.5$ , and its partner at height $-y$ in the bottom lane has the same distance, so this band closes with no reversal and has two sides and two edges. The areas add up, since

\begin{aligned} 30 \cdot 1 + 60 \cdot 1 &= 90 \\[0.4em] &= 30 \cdot 3. \end{aligned}

In space the two pieces are linked together. Moving the cut toward the center line makes the middle Möbius strip narrower, and the center cut is the case where it has no width left, which is why that cut leaves only the long band.