1 Hogwarts Owl Letter Delivery (OLD) Metric

Transcription

1 Berkele Math Circle Kelli Talaska, September 0 STRANGE DISTANCE FUNCTIONS AND STRANGER CIRCLES, PART II Last week we studied taicab distance on a square grid of roads. Toda we will eplore some more distance functions in the plane. We will ask the same kinds of questions as last time. Hogwarts Owl Letter Deliver (OLD) Metric Suppose that magical owls delivering letters must alwas stop off at the Owl Post Office in Hogsmeade. Thus, if Harr wants to send an owl to Ron, his owl must first fl to the Owl Post Office, and then fl to Ron. (It cannot fl straight from Harr to Ron without checking in at the Post Office.) As owls are smart and can fl, the will take straight line paths for each segment the fl. Also, the owls will refuse to let ou mail a letter to ourself; instead the insist ou just hand it to ourself. (Consider all points in the plane this time, not just the grid of roads, and assume the Owl Post Office is at the origin and each block is km b km.) An questions?. On the first grid below, plot Hermione at (, ) and Dobb at (, ). On the second grid below, plot Hermione at (, ), Ron at (, ), Snape at (0, ), Malfo at (, 0) and Dobb at (, ).. Suppose Hermione needs to send an urgent letter b owl to Dobb. Is there onl one shortest path her owl can take, or are there several? How far must her owl fl on the journe? This will be the OLD distance.. Which character above is closest to Hermione b the OLD distance (i.e. the shortest distance the owl can fl, including the stop at the post office)? Make a guess just b looking at the figure, and then check the actual computations to justif or fi our answer.

2 Berkele Math Circle Kelli Talaska, September 0. Let s think about the triangle inequalit now. Can ou find three points A, B and C so that the OLD distance from A to B plus the OLD distance from B to C EQUALS the OLD distance from A to C? Show an eample on the left grid below. (You can denote this AB OLD + BC OLD = AC OLD if ou like.) (Bonus HW: describe ALL such pairs of points, and eplain wh ou have them all.). Time for circles. Let s start with OLD circles around the origin, AKA Owl Post Office. Sketch the points which are OLD distance from the origin, or in other words, points the owl can reach b fling a distance of km. Repeat for km, km, etc.. That seemed too good to be true. Let s focus on Hermione again, on the left grid below. Can ou find an points which are OLD distance from Hermione? OLD distance? Mabe,,,?. From our previous work, can we sketch some OLD circles centered at Hermione? What is weird about these circles?

3 Berkele Math Circle Kelli Talaska, September 0 8. Recall that Hermione is at (, ), Malfo is at (, 0), and Dobb is at (, ). Are there an points which are OLD equidistant from Hermione and Malfo? Are there an points which are OLD equidistant from Malfo and Dobb? Use the grids below to eperiment. 9. If A, B, C, and D are four points such that AB and CD are equal Euclidean distances, must it also be true that AB OLD = CD OLD? If es, eplain wh. If no, show an eample below on the left. 0. If A, B, C, and D are four points such that AB OLD and CD OLD are equal Owl Letter Deliver distances, must it also be true that AB = CD in Euclidean distance? If es, eplain wh. If no, show an eample above on the right.

4 Berkele Math Circle Kelli Talaska, September 0 Teleportation metric Imagine we have invented a ver ecellent portable teleporter, and we can teleport from an starting point to an ending point in one move. There is one eception ou cannot teleport from an point to the same point; that is a waste of a precious teleporter. Thus, the distance from a point to itself is 0 (no moves), and the distance between an two different points is move. An questions about the rules?. Choose a points K (other than the origin) on the graph below. If P is an point, let KP denote the teleporter distance from K to P. Which points are on the teleporter circle of radius around K? (In other words, which points are teleporter move awa from K?) Graph this set, and also describe it in words.. Which points are on the teleporter circle of radius around K? Radius? Radius? You can use the etra graph above if necessar.. Does the teleporter distance satisf the Triangle Inequalit? I.e. is AB + BC AC, no matter what points A, B, and C ou use? When can ou get equalit?

5 Berkele Math Circle Kelli Talaska, September 0. Now, some rascal has tinkered with our teleporter, and it can no longer teleport us ridiculousl long distances. It can onl teleport us a distance of blocks or less in one move. Let KP denote the weak teleporter distance between K and P. Now tr the circles again. Find the weak teleporter circle of radius centered at K on the first graph below, and the weak teleporter circle of radius centered at K on the net graph below. For radius, these are points we can reach with two teleporter moves, but not with onl one move. (You ma change our point K if our first choice is inconvenient now.). Does the Triangle Inequalit seem to work for the weak teleporter? I.e. is AB +BC AC, no matter which points we choose? Below are grids for eperimenting.

6 Berkele Math Circle Kelli Talaska, September 0 Below are some additional metrics we can stud. Come up with our own questions to ask about them. If there is time, ou can start working on them now. If ou have a reall good one, let me know so that perhaps I can share it with the other students net time. Chess King metric Here we imagine an infinite chessboard, represented b an integer grid of points (possible positions for the king). The distance between two points A and B is the number of moves it would take a chess king to move from A to B. An questions about the rules? Questions to eplore (totall fine to list more than ):.. 8. Infinitel Long and Tall Hotel with Onl One Elevator metric. In the Infinitel Long and Tall Hotel with Onl One Elevator, there are infinitel man floors (floor 0, floor, floor, floor, etc.), just one infinitel tall elevator at one end of the hotel, and infinitel man rooms on each floor (room closest to the elevator, then room, then room, and so on). The elevator is the onl wa to move between different floors no stairs (eep, fire safet!), no climbing around like monkes, no drilling holes in the building, and so on. Onl the elevator! It takes one move to go up or down one floor on the elevator, and it also takes one move to walk on a given floor from one room to one that is right net to it. Room on each floor is move from the elevator. An questions about the rules? Questions to eplore (totall fine to list more than ):