Now let us see the situation at infinity on Mobius strip. To do so, we use the hemisphere which we previously used in chapter 31.

Look at Fig. 1.

[1]: The red spot is the center of hemisphere

[2]: It is the cross section passing three points

Look at Fig. 2.

[1]: The red spots on the edge (line at infinity) of hemisphere

[2]: It is a rectangular strip taken from hemisphere

[3]: The right end of [2] is twisted by 180 while the left is fixed.

[4]: It is a view from top of [3].

[5]: We jointed the end lines of [3] and got Mobius strip. The line at infinity is the joint line of Mobius strip, and point

[6]: We newly cut Mobius strip at point

Thus we caught how a straight line that extends to infinity makes a loop at infinity. And the red "straight line" on Mobius strip that is curved surface keeps its character as a straight line.

No one knows whether the straight line acts like this at infinity or not. However, we feel nothing illogical. Naturally, It is based on an idea that there actually exists what is called infinity and a straight line can be stretched to that infinity.

Look at Fig. 3.

What happens if an object that passes line at infinity has some volume?

[1]: We draw a green straight line on Mobius strip of skeleton. And we thread red semitransparent beads with the green line like a necklace. Every gap of balls, center to center in the figure, is equall for our eyes. But we can move it. The bead becomes smaller and smaller when it gets close to the line at infinity. The bead on the line at infinity has no size but it still exists there. It gets bigger again after passing the line at infinity. And it will be back in the original size when it arrives at the start point. Thus the size of any solid body on Mobius strip with the line at infinity changes its own size in our sight. It makes no difference wherever we set the start poin.

[2]: Ball

Look at Fig. 4.

This time we draw paralle lines.

[1]: We cut the hemisphere. The red and blue lines are parallel. They are drawn to sandwich point

[2]: The parallel lines on Mobius strip. Points

[3]: It is the rectangular strip took from Mobius strip. It shall make sense if we regard the both end lines are identical in twist.

Look at Fig. 5.

Two pairs of parallel lines now.

[1]: We have to cut the hemisphere so widely that two pairs of parallel lines are contained. On the ground the red parallel lines are just under the hemisphere but the blue parallel lines are off to the side.

[2]: It is the bird's-eye view of [1].

[3]: It is Mobius strip made from [2]. It is drawn to show both of the line at infinity and the area where the two pairs cross each other are facing to us at a time. On the line at infinity, the red parallel lines meet at point

Look at Fig. 6.

[1]: We shrunk and cut Mobius strip in Fig. 5, and got the rectangular strip. The central black line is line at infinity. Both end lines are at

[2]: The top right area of Mobius strip in Fig. 5 is not clearly seen. So we drew it with line at infinity in perpendicular to us since we have uderstood around line at infinity. As colored in green and pink, there appears two quadrilaterals. But the pink one is not easy to see as a quadrilateral, isn't it?

[3]: We shifted line at infinity of [1] a little so that the two quadrilaterals are in the rectangular strip without dividing either of the two into parts. Any two pairs of parallel lines can be drawn like this. On an ordinary plane two pairs of parallel lines make one parallelogram. On Mobius strip we find the pink quadrilateral besides the green one. The pink quadrilateral is infinitely large and is laide across line at infinity.

[4]: Let us change our view point.

We agree for the present that two straight lines that meet each other at infinity are parallel. On that basis, we pay attention to the red straight line

The rectangular strip is the same as [3]. The straight lines

shows the new situation with new line at infinity which is located in the middle.

Well, is it logical? What in un-paralle became what in parallel! However, it is true. They say "It is a character of what is called line at infinity." Wait a minute. It's funny for us now. Let us think of it in course of time.

[5]: The colored quadrilaterals in [3] are taken out together with their around. is of the green quadrilateral and is of the pink. The domains

Look at Fig. 7.

We deform Mobius strip and try to see relations of

[1]: We cut the hemisphere in X-shape so that both the red line and the blue line are just in the cutout as shown. is the cutout.

[2]: We twist the X-shaped rectangular strip and joint the end lines as shown. It is the same as two Mobius strips glued back to back. The two black lines

[3]: It is colored corresponding with [5] of Fig. 6. The domains

Let us see what differences between a paper-made Mobius strip and that of the formula are.

Look at Fig. 8.

[1]: It is one of appearances of a paper-made Mobius strip. We can see how it is twisted, but it is not very useful to see things on the surface. First of all, it is not easy to set up coordinate axis. In addition to that, there are some other reasns why we feel difficulty eith paper-made Mobius strip.

[2]: Turn [1] a little, then we can see the twisted part clearly. The gray regular nick lines are fairly perpendicular to the whole edge.

[3]: We turned [1] more. Now we can see that the strip is twisted three times. Two of twists are in the same direction and one in the opposite direction, and totally a single twist in net. This Mobius strip hides the surface so much with extra twists. That is why the paper-made Mobius strip is no good for observation.

Look at Fig. 9.

Let us see relations between a rectangular strip and Mobius strip made with the formula.

[1]: It is in case that the length of edges of the rectangular strip is identical to that of Mobius strip.

We nick the rectangular strip into

The nick lines are broken at the center line. They shall be smoothly curved if we draw them accurately. But we omitted on purpose for attract attention. Compared with the rectangular strip.

, the figure is enlarged.

It is a bird's-eye view.

It is the view around points

Points on the edges and center line of the rectangular strip and Mobius strip correspond each other. But the length of their nick lines are quite different except nick

[2]: This time, first we draw Mobius strip, and then we cut and open it as a rectangular strip.

The green center line is a circle. Its center is

It is a bird's-eye view. is the angle of nicks. See Fig. 3. In [1] lengths of the red edge and black edge are the same as well as the rectangular strip. But in [2] it isn't. In the figure the length of black line is one and a half times as long as the red line.

It is the view around points

It is what we cut and opened Mobius strip while keeping the length of edge with nicks. It is formed into a trapezoid and the length and slope of nick lines are distorted.

[3]: It is Mobius strip made of a rectangular strip sheet of paper. Its edge is dyed in red and black correspondingly to the rectangular strip. Just for a nice look, the number of nick lines is increased.

Naturally, all nick lines are equal in length and they are fairly perpendicular to the edge. And the length of red and black edges are equal, too.

We turned Mobius strip so that we can see around nicks lines

Points on Mobius strip and a flat belt can be correponded to each other by one-to-one mapping. It has nothing to do with the shape of Mobius strip or the belt. However, their partial density or like of points is never be equal. Paper-made Mobius strip also has some expansion and contraction though we can not clearly see it with our own eyes.

Let us name the red edge

Look at Fig. 10.

[1]: Let us distinguish edges of a belt as shown.

[2]: We bend an end of the belt and true up the edges of them in flat. And we push the bended belt and glue touched portion.

[3]: Then, we get a loop with a split. The dotted line (*) is the joint. It is called Split-Wing Loop, though it does not appear in a mathematical book.

Split-Wing Loop does not look twisted anywhere. But it has neither the front nor the back, that is one-sided. You will see it by following the surface. Is it really twisted?

[4]: Forld Split-Wing Loop, and glue the yellow side

[5]: We turn [4] and expand its hole to see how twisted. The width is shrunk a bit for drawing. It is twisted only one time as we expected. When we cut the joint (*) and open the hole wide, we get an ordinary Mobius strip.

[6]: It is only the edge of Mobius strip. It is colored similarly to Split-Wing Loop of [3]. Obviously, the edge of Mobius strip is the same as that of Split-Wing Loop. Namely Split-Wing Loop is twisted. Then, how on earth, what was actually done in the process from [3] to [4]?

[7]: Let us see the trick.

Let us begin with [2]. We slide the side

Turn over the face with the side

[8]: The side

Mobius strip can be made by cutting a torus. So let us try some different cuts.

Look at Fig. 11.

[1]: Sometimes it is called a twisted triangular prism. A single surface is twisted and a hollow is formed. The ridgeline is single, too. The surface looks similar to that of Mobius strip but it is not so. The inside and outside are the front and back respectively. We can see only front side if the inside is the back. We can see only one side. That is why we feel something like Mobius strip.

[2]: The cross-section of loop is a regular triangle. This triangle turns as , and every circuit by 120 each, and gets back to the original location by three circuits.

Look at Fig. 12.

[1]: Let us see vertices of the twisted triangular prism in a solid torus. The red line is the locus of the cut. It travels three circuits. The dotted red line is on the invisible back side. The dull yellow ring is a cross-section of the tube. The black spot

The red curve makes 3 circuits with 1 turn. Say, 3 circuits 1 turn. With this wording, the simplest Mobius strip is 2 circuits 1 turn.

[2]: As a trial we made 2 circuits 5 turns. Point

[3]: It is the bird's-eye view of Mobius strip taken out of [2]. The dotted lines are parts where were beyond the tube of torus.

[4]: The width of Mobius strip is limited. It is inconvenient for us. How about a torus with twisted surface? Red lines indicate how twisted. It seems convenient. But, first of all, the torus has front and back surfaces. It never change whatever twisted. Do not forget that we sometimes see torus from its inside.

Look at Fig. 13.

[1]: It is twisted 90 a circuit. Its section is a square. You may call it a twisted quadragular prism.

[2]: It also is twisted 90 a circuit. But the surface cuts itself orthogonally all the way. The section is cross-shaped. But the black frame is the same as that of [1]. The surface twists 180 by two circuits, but we are bewildered if it can be called Mobius strip or not, aren't we?

[3]: It is a transparent view of [2]. The color of the second circuit is changed for easy observation.

[4]: Gears in a row are on the surface of [2]. It is copied from Mobius Strip in Mathworld Wolfram. You should click and visit the web site. You shall be impressed by moving gears. And you must feel what function a twisted surface has.

Also there is a moving Mobius strip in A HREF="http://www.palmyra.demon.co.uk/illusion/geometry/geometry.htm">Mobius animation. M.C. Escher drew Mobius strip with ants on it. Those ants are actually crawling in RUBAN DE MOBIUS.

Well, there are lots of things about Mobiusu strip. But let us go back to projective geometry anyway.

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