Something you can do with a Sheet of Paper
Hey look, it's math even normal people can do!
I actually thought about this while doing an assignment for geometry class, which involved drawing a figure on graphing paper, of which I had none at the moment. I'm not poor, and it's probably unusual for a math geek such as myself to not have graphing paper, although I guess I don't have any because I don't believe in measuring stuff in geometry. It's the theory that counts. So when you draw a function, for example, it's not exactly important to know exactly where every point is, only that you know basically what it looks like and its properties. I prefer working with general cases that work on lots of things rather than only a few specific things.
But I digress. Anyways, not having any graphing paper, and not wanting to lose any marks for not drawing the picture, I decided to still draw it; at least I'll get something for knowing what the picture looks like. Even more unusual to some people is that I don't use rulers, compasses or other aids for geometry. Well, the reason I don't is for the same reason as above--the general picture is sufficient enough; not the exact picture.
Anyways (again) here is what we were asked to do:
"Take a sheet of graph paper, draw a horizontal segment of length 11 (which occupies roughly the middle third of the sheet), draw a line through the left endpoint A of the segment with the slope of -45 degrees, draw another line through the right endpoint B of the segment with the slope of 45 degrees. Draw a series of lines through the following pairs of points: pick an integer x between -9 and 9, move the point A along the first line by x squares and get a new point A(x), move the point B along the second line by x squares and get a new point B(x); (for positive x A(x) and B(x) lie to the right of A and B respectively); and then draw a line through A(x) and B(x). The resulting picture will clearly show a conic section (parabola) that is tangent to all lines that you have drawn."
So I didn't have any rulers, no other geometrical aids and no graphing paper--how did I accomplish this?
I actually had a lot of room left on the side of the page I was writing the solution on, so it was easy to do this, but for those who want to try this, you can take any blank sheet of paper you want--you don't need any rulers, compasses, geometrical aids or graphing paper to do this--YOU DON'T EVEN NEED TO WRITE ANYTHING AT ALL! The only reason I did was that I was afraid the person marking my assignment wouldn't notice it. Now, onto the procedure itself:
1) Fold your sheet of paper diagonally both ways so that you now have an X across one side of your sheet. Don't worry if your sheet of paper isn't square; just fold the diagonal over so that the corner touches the opposite edge of the sheet and the other corner is folded exactly in half. Now make sure your sheet is completely unfolded.
2) I did this on an 8.5 x 11, presumably, so a third of the page is approximately 2.8 in., but the "X" on my page only left 2.5 in. of extra space above it, but it was pretty close, so I used that as a guide. I will be referring to "horizontal" as the width of the page that's the shortest and "vertical" as the length of the page that's the longest--so your page should be facing you from a "portrait" perspective (see the page setup command under your browser to see what I mean) If your page is the same dimension as well, simply fold the sheet horizontally so that there's a crease across where the edge of the page meets both ends of the "X". Then fold the sheet in half again (in the opposite direction of the first fold) so that there's another horizontal crease somewhere before the middle of the "X". On my sheet of paper, the second horizontal crease would be approximately 5 in. down from the side of the page farther from the "X". Once again, unfold your sheet completely.
3) To avoid confusion, it would be best to mark the second horizontal crease now with a pencil or pen. Marking down creases with pencil is just about as easy as using a ruler to draw straight lines. Of course, there are other ways of drawing straight lines without rulers, including using the edge of the page as a straightedge. I used to do this a lot in high school. Now make any horizontal crease (fold it in the direction opposite of the second horizontal crease) you want between the two horizontal creases from the second step. Now fold the page over the second horizontal crease (but keep the new crease you made folded). Now fold a new horizontal crease on the "X" where your new fold meets the rest of the page. Now unfold the page.
4) Look for two points in which two of your creases meet each other:
-one point is between the second to last horizontal crease you made and one of the diagonals you made at the beginning of the procedure.
-the other point is between the last horizontal crease you made and the other diagonal.
Now fold a crease between those two points. You may repeat this step with two other points by switching the diagonals.
Repeat steps 3 and 4 until you can start to see the conic.
Unfortunately, I haven't gotten my assignment back yet, so I can't scan it and show it to people. However, I did take a picture of this one so that it will be easier to follow the instructions. This is actually the back side of the page, which I took the picture of because the creases show up better, and you can actually see the conic better (it's a parabola above the deep crease in the middle). The horizontal segment from the problem itself is the one in the middle which curves the page differently from the other horizontal creases.
How this works (there are some math proofs ahead, so you may want to skip them if you don't get it)
1) The diagonals:
I have never seen too many sheets of paper that didn't have 90 degree corners--most conventional sheets of paper are rectangular or square, and therefore, have 90 degree corners. Any bisector of such a corner must be 45 degrees, which are the 45 degree lines on the endpoints A and B in the assignment. I admit I made the diagonals before I made the horizontal line segment, although it doesn't really change the fact that you'll still end up with a conic.
2) The two horizontal creases:
The first one was made to mark out where 2.5in. of the page would be, and since the diagonals are 45 degrees with the edge of the page, the second horizontal crease and the first one form two sides of a square (a square has diagonals at 45 degrees). This would mean that the diagonal crosses the second horizontal crease 2.5 in. into the page on both sides--the remaining section is roughly a third of 8.5 in, although it would actually be a bit bigger, since I didn't have a ruler to measure out where 2.8 in. should be. In either case, it doesn't really matter, but I wanted the line to be as close as possible to what the assignment asked, but for your own purposes, it won't be necessary. You can place the line above the intersection of the diagonals wherever you want. Your resulting conic will just look slightly compressed or dilated.
3) The other horizontal creases:
I just used these as guidelines for the tangents. Since the diagonals are both at 45 degrees to the bottom of the page, making two creases both parallel with the bottom of the page and the same vertical distance away from the second horizontal crease makes the diagonals between them the same distance. They are, afterall, forming the same squares if you draw vertical lines between the intersections of the diagonals with one crease to the second horizontal crease, and one from the intersection of the diagonals with the second horizontal crease to the other crease--you'll end up with four boxes, all the same size. This is the same as choosing an x and the points A(x) and B(x) the same distance away from the second horizontal line.
4) The two points and the crease between them:
This is the supposed tangent of the conic--we've already calculated where the points should be from the guidelines above, so all we need to do is just make them.
Math can be fun and made accessible to people.
I actually thought about this while doing an assignment for geometry class, which involved drawing a figure on graphing paper, of which I had none at the moment. I'm not poor, and it's probably unusual for a math geek such as myself to not have graphing paper, although I guess I don't have any because I don't believe in measuring stuff in geometry. It's the theory that counts. So when you draw a function, for example, it's not exactly important to know exactly where every point is, only that you know basically what it looks like and its properties. I prefer working with general cases that work on lots of things rather than only a few specific things.
But I digress. Anyways, not having any graphing paper, and not wanting to lose any marks for not drawing the picture, I decided to still draw it; at least I'll get something for knowing what the picture looks like. Even more unusual to some people is that I don't use rulers, compasses or other aids for geometry. Well, the reason I don't is for the same reason as above--the general picture is sufficient enough; not the exact picture.
Anyways (again) here is what we were asked to do:
"Take a sheet of graph paper, draw a horizontal segment of length 11 (which occupies roughly the middle third of the sheet), draw a line through the left endpoint A of the segment with the slope of -45 degrees, draw another line through the right endpoint B of the segment with the slope of 45 degrees. Draw a series of lines through the following pairs of points: pick an integer x between -9 and 9, move the point A along the first line by x squares and get a new point A(x), move the point B along the second line by x squares and get a new point B(x); (for positive x A(x) and B(x) lie to the right of A and B respectively); and then draw a line through A(x) and B(x). The resulting picture will clearly show a conic section (parabola) that is tangent to all lines that you have drawn."
So I didn't have any rulers, no other geometrical aids and no graphing paper--how did I accomplish this?
I actually had a lot of room left on the side of the page I was writing the solution on, so it was easy to do this, but for those who want to try this, you can take any blank sheet of paper you want--you don't need any rulers, compasses, geometrical aids or graphing paper to do this--YOU DON'T EVEN NEED TO WRITE ANYTHING AT ALL! The only reason I did was that I was afraid the person marking my assignment wouldn't notice it. Now, onto the procedure itself:
1) Fold your sheet of paper diagonally both ways so that you now have an X across one side of your sheet. Don't worry if your sheet of paper isn't square; just fold the diagonal over so that the corner touches the opposite edge of the sheet and the other corner is folded exactly in half. Now make sure your sheet is completely unfolded.
2) I did this on an 8.5 x 11, presumably, so a third of the page is approximately 2.8 in., but the "X" on my page only left 2.5 in. of extra space above it, but it was pretty close, so I used that as a guide. I will be referring to "horizontal" as the width of the page that's the shortest and "vertical" as the length of the page that's the longest--so your page should be facing you from a "portrait" perspective (see the page setup command under your browser to see what I mean) If your page is the same dimension as well, simply fold the sheet horizontally so that there's a crease across where the edge of the page meets both ends of the "X". Then fold the sheet in half again (in the opposite direction of the first fold) so that there's another horizontal crease somewhere before the middle of the "X". On my sheet of paper, the second horizontal crease would be approximately 5 in. down from the side of the page farther from the "X". Once again, unfold your sheet completely.
3) To avoid confusion, it would be best to mark the second horizontal crease now with a pencil or pen. Marking down creases with pencil is just about as easy as using a ruler to draw straight lines. Of course, there are other ways of drawing straight lines without rulers, including using the edge of the page as a straightedge. I used to do this a lot in high school. Now make any horizontal crease (fold it in the direction opposite of the second horizontal crease) you want between the two horizontal creases from the second step. Now fold the page over the second horizontal crease (but keep the new crease you made folded). Now fold a new horizontal crease on the "X" where your new fold meets the rest of the page. Now unfold the page.
4) Look for two points in which two of your creases meet each other:
-one point is between the second to last horizontal crease you made and one of the diagonals you made at the beginning of the procedure.
-the other point is between the last horizontal crease you made and the other diagonal.
Now fold a crease between those two points. You may repeat this step with two other points by switching the diagonals.
Repeat steps 3 and 4 until you can start to see the conic.
Unfortunately, I haven't gotten my assignment back yet, so I can't scan it and show it to people. However, I did take a picture of this one so that it will be easier to follow the instructions. This is actually the back side of the page, which I took the picture of because the creases show up better, and you can actually see the conic better (it's a parabola above the deep crease in the middle). The horizontal segment from the problem itself is the one in the middle which curves the page differently from the other horizontal creases.
How this works (there are some math proofs ahead, so you may want to skip them if you don't get it)
1) The diagonals:
I have never seen too many sheets of paper that didn't have 90 degree corners--most conventional sheets of paper are rectangular or square, and therefore, have 90 degree corners. Any bisector of such a corner must be 45 degrees, which are the 45 degree lines on the endpoints A and B in the assignment. I admit I made the diagonals before I made the horizontal line segment, although it doesn't really change the fact that you'll still end up with a conic.
2) The two horizontal creases:
The first one was made to mark out where 2.5in. of the page would be, and since the diagonals are 45 degrees with the edge of the page, the second horizontal crease and the first one form two sides of a square (a square has diagonals at 45 degrees). This would mean that the diagonal crosses the second horizontal crease 2.5 in. into the page on both sides--the remaining section is roughly a third of 8.5 in, although it would actually be a bit bigger, since I didn't have a ruler to measure out where 2.8 in. should be. In either case, it doesn't really matter, but I wanted the line to be as close as possible to what the assignment asked, but for your own purposes, it won't be necessary. You can place the line above the intersection of the diagonals wherever you want. Your resulting conic will just look slightly compressed or dilated.
3) The other horizontal creases:
I just used these as guidelines for the tangents. Since the diagonals are both at 45 degrees to the bottom of the page, making two creases both parallel with the bottom of the page and the same vertical distance away from the second horizontal crease makes the diagonals between them the same distance. They are, afterall, forming the same squares if you draw vertical lines between the intersections of the diagonals with one crease to the second horizontal crease, and one from the intersection of the diagonals with the second horizontal crease to the other crease--you'll end up with four boxes, all the same size. This is the same as choosing an x and the points A(x) and B(x) the same distance away from the second horizontal line.
4) The two points and the crease between them:
This is the supposed tangent of the conic--we've already calculated where the points should be from the guidelines above, so all we need to do is just make them.
Math can be fun and made accessible to people.