MÖBIUS STRIP
Surface studied by Listing and Möbius in 1858. August Ferdinand Möbius (17901868): German astronomer and mathematician. Other name: Möbius (or Moebius) band, ring, belt. 
A Möbius strip is a surface obtained by sewing together two sides of a rectangular strip with a halftwist, or any topologically equivalent surface.
Maple program giving an animation of the opposite construction. with(plots):a:=1/2:b:=1/3:c:=1/6:d:=2/3:e:=1/3:C:=4/5: x0:=(1+d^2*t^2+2*d*e*t^4+e^2*t^6)/2:x:=(a*t+b*t^3+c*t^5)/x0:y:=(d*t+e*t^3)/x0: z:=C/x0:t:=tan(tt): a1:=diff(v1,tt):a2:=diff(v2,tt):a3:=diff(v3,tt): v1:=diff(x,tt):v2:=diff(y,tt):v3:=diff(z,tt): b1:=v2*a3a2*v3:b2:=a1*v3v1*a3:b3:=v1*a2a1*v2: n1:=simplify(v2*b3b2*v3):n2:=simplify(b1*v3v1*b3):n3:=simplify(v1*b2b1*v2): dn1:=diff(n1,tt):dn2:=diff(n2,tt):dn3:=diff(n3,tt): c1:=n2*dn3dn2*n3:c2:=dn1*n3n1*dn3:c3:=n1*dn2dn1*n2: facteur:=simplify(sqrt(b1^2+b2^2+b3^2)/(b1*c1+b2*c2+b3*c3)): c1:=simplify(c1*facteur):c2:=simplify(c2*facteur):c3:=simplify(c3*facteur): ds:=simplify(sqrt(v1^2+v2^2+v3^2)): s:=a>evalf(Int(ds,tt=0..a,4))/4: d:=a>plot3d([x/s(a)+u*c1/s(a),y/s(a)+u*c2/s(a),(z+2*C)/s(a)+u*c3/s(a)],tt=a..a,u=1/3*s(a)..1/3*s(a),grid=[150,2],style=patchnogrid): n:=40:display([seq(d(k*Pi/2.0001/n,50),k=1..n)],orientation=[60,80],lightmodel=light2,insequence=true); 
Therefore, we obtain a Möbius strip by turning regularly a segment of a line with constant length around a circle with a halftwist or, more generally, an odd number of halftwits; these various strips are homeomorphic, but not isotopic in (it is not possible to pass continuously from one surface to the other), and for each number of halftwists, there exist two isotopy classes, mirror images of one another:
1 halftwist  3 halftwists  9 halftwists  
righthanded strip  
lefthanded strip 
Moreover, the boundary of the strip with 3 halftwists is a trefoil knot, and, more generally, the boundary of the strip with 2p + 1 halftwists is a toroidal knot of order (2p + 1, 2).
These rotoidal surfaces are not developable: they cannot be obtained from a piece of paper without tearing; inversely, the strip that is naturally obtained with a rectangular piece of paper does not have a simple parametrization; here is one owed to W. Wunderlich (cf. above), in the righthanded case with one halftwist:
The MöbiusWunderlich strip has the property that it develops into a rectangle and it minimizes at all points the strain energy. 
Here is another example of a Möbius strip developable into a rectangle, composed of 3 cylindrical sections joined by planar sections:
Lefthanded strip with one halftwist: the radius of one of the cylinders is equal to twice that of the other ones and the planar sections are parallel.  Righthanded strip with 3 halftwists. (Production: Alain Esculier) 
Here are two other developable Möbius strips obtained by putting strips cut on cones next to one another.
The blue and green strips are cut on cones of revolution; the red strip on two cones of revolution; the pattern is unfortunately not a rectangle, but a parallelogram, hence the connection with a right angle. 
This is a Möbius strip with 3 halftwists instead of 1, obtained thanks to 3 conical strips, the directrix of the cones being Viviani's curve. The boundary is selfparallel, but the pattern does not have linear edges. 
Möbius Strip Hearts
If you thought the heart was the most romantic shape, then it’s time to meet the Möbius strip. When two of these famous mathematical surfaces are placed together, you’ll end up with two interlocking hearts. Nothing says “I love you” better than a little mathematical wonder!
All you need to make this mathematical statement of love is paper, scissors, and tape. That’s it!
What is a Möbius Strip?
The easiest way to explain a Möbius strip is to make one.
We first cut a strip of paper about 11.5 inches by 810 inches.
Next, we made a loop with the strip, put a half twist in one side (turned one end 180 degrees), then taped it together.
Now to see what makes the Möbius strip famous, we drew a line down each “side” without lifting our pen. The Möbius strip is famous because it only has one side and one surface!
Now that we mastered the Möbius strip, we challenged ourself by making two Möbius strips into two interlocking hearts.
Making Möbius Hearts
For the next step, we grabbed two strips of paper.
For the first strip, we made the half twist to the left. Then we taped down both sides of the strip.
We made the second Möbius strip by making the half twist to the right. It’s key that each strip was made by twisting the strip in opposite directions.
We took the two Möbius strips and placed them together at a 90 degree angle where they were taped. We taped the 2 strips together on both sides.
Starting with one strip, we gently folded it in half and cut a small slit in it.
Next, we cut the strip in half lengthwise.
We kept cutting right through the intersection of the next Möbius strip.
Once we finished cutting the first strip into two, we finished cutting the second strip (the purple strip) in half, lengthwise,
Voila! We had two interlocking Möbius hearts!
More about the Möbius Strip
The Möbius strip is named after mathematician August Ferdinand Möbius, who came up with the idea in 1858.
Interestingly, German mathematician Johann Benedict Listing developed the same idea a few months earlier but the strip was named after Möbius.
mobiusdissection
Mobius Dissection
Visualize whirled peas.
 Introduction
 Cutting a Mobius strip and its relatives provides
an opportunity to visualize what will happen, make hypotheses, and to
be surprised by the results of an experiment.  Material
 Several strips of paper ( a good size is 2.5 cm wide and 25 cm long)
 if you use brown packing tape with adhesive on one side you can skip the tape below.
 tape
 scissors
 a felt tip pen
 a paper clip
You'll need to master the techniques for counting
sides of a loop of paper by drawing a line around its center, and for
counting halftwists using a paper clip that were described in the
Mobius
exploration before you begin this one.
To Do and Notice/ What's Going
On?
Bend a paper strip into a loop with no twists.
Make a line down the middle of your loop.
Visualize what you will get when you cut this loop along the line.
 Cut the loop and observe what happens.
 Congratulations.
 Getting twisted.
 Give the paper a half twist and tape or glue the
ends together to make a Mobius strip.  Mark a line down the center of this
strip.  Visualize what you will get when you cut the
Mobius strip along the line.
Perhaps you thought that you would get two Mobius
strips. Surprise!
What you get is described in this
limerick:
A mathematician confided That a Mobius band is onesided, And you'll get quite a laugh, If you cut one in half,
For it stays in one piece when divided.
A hoop and a Mobius strip cut in half.
How many sides does the resulting band have? Draw
a line to find out.
How many halftwists. Count them with the twist
test to find out.
Half a Mobius strip. marked down the middle and taped down for
counting twists.
Now cut the resulting band in half.
Another surprise.
Count the number of sides each of the resulting bands
possesses.
A half a Mobius strip cut in half again.
Next cut a new Mobius strip onethird of the way
in from one edge. The cut will meet itself eventually.
You end up with two loops, you've cut the edge off
the original Mobius strip leaving a thinner version of itself intact
at the center, and producing a second loop, twice as long, with two
halftwists.
Count the sides of, and twists in, the resulting rings.
Try cutting a slice off the Mobius strip 1/4 or
1/5 or some other distance in from the edge. Predict what you expect
to happen then do the experiment.
Try slicing strips with 3 halftwists or other
numbers of half twists. Predict the answer you expect then do the
experiment.
Can you find the rule that tells you how many
twists you get when you bisect a loop with n half twists? Hint, Loops
with even numbers of halftwists follow a rule that is different from
those with odd numbers of halftwists.
 What's Going On?
 When you cut the Mobius strip in half you made a
strip twice as long with 4 half twists.  Cut the halfMobius in half again and you get two
linked strips, both with 4 half twists.  Cut the Mobius strip in thirds and you make two
linked loops one with one half twist and one with 4 half
twists.  Mark a line 1/n from one side of a Mobius band of
width W, then cut along this line.
There will be a strip left in the middle of width
W – W/nW/n.
A strip will be left in the middle for any value of n except n=2.
This is why marking a line down the middle of the
Mobius strip, 1/2 way from one edge, produces a different result from
all other fractions.
The rule
If you bisect a strip with an even number, n, of
half twists you get two loops each with n halftwists. So a loop with
2 halftwists splits into two loops each with 2
halftwists.
If n is odd, you get one loop with 2n + 2
halftwists. So a Mobius strip with 1 halftwist becomes a loop with
2+2 = 4 halftwists.
Scientific Explorations with Paul Doherty  © 2000  20 October 2000 
The Hidden Twist to Making a Möbius Strip
In the field of symplectic geometry, a central issue involves how to count the intersection points of two complicated geometric spaces. This counting question is at the heart of one of the most famous problems in the field, the Arnold conjecture, and it’s also a matter of basic technique: Mathematicians need to know how to make these counts in order to do other kinds of research.
As I describe in my article “A Fight to Fix Geometry’s Foundations,” developing a method for counting these intersection points has been a drawnout and sometimes contentious process.
A reliable, widely understood, errorfree approach has presented a challenge for a number of reasons, from the lack of a shared vocabulary when a new field gets started (symplectic geometry only really took off beginning in the 1990s), to the nature of the problem itself: Simply put, it’s hard.
The difficulty lies in the fact that for subtle reasons, it’s not possible to count the intersection points all at once.
Instead, mathematicians need to break the space down into “local” regions, count intersection points in each region, and add those together to get the “global” count.
Piecing together local counts has proved to be a more delicate and technically demanding task than mathematicians realized at first: If you’re not careful about how you draw your local regions, you could easily omit one intersection point or doublecount another.
The following illustrations explore the difficulty of the task using a Möbius strip (a twodimensional circular band with a twist in it). The Möbius strip has two circles passing through its surface.
The question is: How many times do the two circles intersect each other? As you’ll see, the answer seems to be one thing when you look at the strip all at once, and another if you’re not careful when you cut the Möbius strip into two pieces.
A Counting Puzzle
Mathematicians want to count intersection points, but certain obstacles prevent them from counting all those points directly. To overcome those obstacles, they divide the manifold into bitesized “local’’ regions, count the intersections in each and add those together to get a count for the whole manifold.
However, if mathematicians are not careful about how they combine counts from local regions, they can easily end up with the wrong count for the whole manifold. The delicacy of adding local counts together is evident in this simple example.
Möbius Rip
Take a Möbius strip. Draw two circles running through it. If you look at the whole Möbius strip, the two circles have to intersect each other at least once: One circle starts above the other, but ends up below it because of the twisting nature of the strip.
Now cut that same Möbius strip into two pieces. The cuts remove the twist in the strip. Draw two circle segments on each piece. Without the twist, it’s easy to draw the circle segments so that they run parallel to each other and never intersect.
As a consequence, you might erroneously conclude that the number of intersections on the whole Möbius strip is zero.
Mathematicians in symplectic geometry have learned that gluing together “local” pieces to recover a “global” intersection count is a much more complex process than they first imagined.
How to Make a Mobius Strip
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1
Cut a strip of paper about 6 inches (15 cm) long and 1.5 inches (3.8 cm) in width. When cutting the strip, the measurements do not need to be accurate, these are just suggested. Try to keep the width even so that you have a long thin rectangle.
 To simplify the process, you can simply cut down the edge of a plain sheet of paper to make your strip.

2
Label the corners of the strip A, B, C, and D. In the top left corner of the strip, write a small letter A; in the top right corner, write a small B; the bottom left, the letter C; and the bottom right, the letter D. You will use these letters to align the strip after the twist.
 The size you make the letters doesn't matter, but the position of each one is important for the twist step.

3
Twist the AC side a half turn and bring it to the BD side. Hold the two ends in your hands, give the AC side of the strip a half twist and join it to the BD side. Match the letters, A to D and B to C and tape the edges together. Once the edges are taped, you have completed the Mobius strip.
 You can twist the paper more than one time and still have a Mobius strip.[2]

1
Draw a line along the middle of the strip. Using a pen or pencil, start at any point in the middle of the strip and draw a line all the way around without lifting your pen.
Eventually, the pen will end up back at the point you started drawing. You have drawn a line on both sides of the loop – but without lifting your pen or crossing any edge.
How did this happen? The paper has only one side!
 Start at a different point in the Mobius strip and see if the same thing happens.

2
Color the edge of the strip with a highlighter. Take a highlighter and start coloring the edge of the Mobius strip without lifting the highlighter from the strip. Continue with the marker until you reach the point at which you started. You'll find both edges are colored. This indicates that the Mobius strip has only one edge!

3
Cut the Mobius strip along the central line you drew earlier. With a pair of scissors, poke a hole into the middle of the Mobius strip and cut along the line until you reach the beginning cut. It does not, as you'd expect, fall apart into two separate loops; instead you now have a single, larger onesided loop![3]

4
Cut the Mobius strip 1/3 of the way away from the edge.
Möbius Strips  Brilliant Math & Science Wiki
 What happens when a Möbius strip is cut down the center line?
 You should first try this experiment yourself!
Instead of getting two strips, the result is a single strip with one full twist (360∘).(360^circ).(360∘). □_square□
 Can you see why this is the case by considering the directed edge identification diagram for the Möbius strip?
Now, what happens if you draw a center line down this resulting figure? Try cutting the strip down the center line a second time and see what you get.
 Can you explain the result?
Now, instead of drawing a line down the center of the Möbius strip, draw a line with distance 13w frac{1}{3}w31w from the edge. What happens when a Möbius strip is cut down this line? Is it the same as the above example?
 You should first try this experiment yourself!
 The result is two strips:
 one Möbius strip of width 13w frac{1}{3}w31w and length lll and
 one longer strip of width 13w frac{1}{3}w31w and length 2l2l2l with one full (360∘)(360^circ)(360∘) twist.
This is different than in the first example because the cut along the middle of a Möbius strip returns to the starting point after the length lll of the strip. However, by cutting from onethird of the way from the edge, the return to the starting point occurs at a total distance of 2l2l2l. □_square□
Can you manipulate the interlocked rings (without creating any sharp creases) to stack on top of each other to form a stack of 3 Möbius strips? (This is shown in Martin Gardner's Mathematical Magic Show.)
Take a piece of paper and make a kkk half twists of 180∘180^circ180∘ before identifying the edges (((where k=1k=1k=1 half twist gives a Möbius strip).).). How many boundaries and how many sides does the resulting figure contain?
You should first try this experiment yourself!
If kkk is even, the result has two boundaries and two edges. A special case is k=0k=0k=0, which gives the cylindrical shell as shown in the edge identification diagram. For kkk even, if the strip is cut along the center line, it will separate into two rings, and each ring will contain kkk half twists.
If kkk is odd, the result has one boundary and one side. If this strip is cut along the center line, the result is a single strip with 2k2k2k half twists. □_square□
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