Inside or outside?
Tonight's idea
Draw a closed loop that never crosses itself. However tangled it is, it splits the paper into two parts: an inside and an outside. (This is the Jordan curve theorem. It sounds obvious, yet a full proof is hard.)
When the loop is tangled, your eye cannot tell which part a dot is in. Counting can.
From the dot, draw a straight path to somewhere clearly outside the whole loop. Count how many times the path crosses the loop. Odd: the dot is inside. Even: the dot is outside.
Why? Walk the path backwards, starting at the far end. You start outside. Every crossing changes your side: outside, inside, outside, inside. After an odd number of crossings you are inside. After an even number you are outside again.
The path from A crosses 5 times, so A is inside. One warning: if your path only brushes the loop without going through, pick another path.
Watch one: dot B
B looks buried in the middle. Go straight down from B and count: 4 crossings. Even, so B is outside.
Try straight up instead: 6 crossings. A different count, but still even. It has to be, because where B is cannot depend on the path you choose. Two paths from one dot give two odd counts or two even counts.
Your turn
- In the loop above, is each of P, Q and R inside or outside? Draw a straight path from each dot and count.
- From S, draw two different straight paths that leave the loop. Write down both counts. What do they have in common? Is S inside or outside?
- No picture needed. (a) A straight path from a dot crosses a loop 7 times. Inside or outside? (b) You draw another straight path from the same dot. Could it cross 4 times? 9 times? 10 times? (c) A straight line from X to Y crosses the loop 5 times, and X is inside. Where is Y?
- A loop is hidden under a book, apart from one straight line drawn right across the page. Both ends of the line are outside the loop. The loop crosses the line 6 times, cutting it into 7 pieces, numbered 1 to 7 from left to right. (a) Which pieces are inside the loop? (b) Explain why a line like this could never be crossed exactly 5 times.
Stretch: two loops
Draw two closed loops on one page. Neither crosses itself, but they cross each other as often as you like. (They must cross properly, not just touch.) Show that the number of points where they cross is always even.
Answers
- P is inside: straight up crosses 3 times, straight down 3. Q is outside: up 4, down 4. R is outside: up 4, right 2. (R's channel opens at the right-hand end.) Your paths may give other counts, but each must be odd for P and even for Q and R.
- S is inside. Straight up crosses 5 times, down 3, right 5, left 9. Every count is odd. Two paths from one dot can differ, but only by an even number.
- (a) Odd, so inside. (b) 9 is possible. 4 and 10 are not: the dot is inside, so every path out crosses an odd number of times. (c) Each crossing changes sides. Five changes starting from inside: out, in, out, in, out. Y is outside.
- (a) Piece 1 is outside. Each crossing changes sides, so the pieces go out, in, out, in, out, in, out. Pieces 2, 4 and 6 are inside. (b) The line starts outside and ends outside. To finish on the side you started, you must change sides an even number of times. Five changes would leave the far end inside.
Stretch. Put your finger on the first loop at a point that is not a crossing, and walk once all the way round it. Each time you cross the second loop you change sides of it. You finish exactly where you began, so on the same side of the second loop as you started. That needs an even number of changes. So the loops cross an even number of times.