Editing 2625: Field Topology
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==Explanation== | ==Explanation== | ||
− | + | {{incomplete|Created by A DONUT, OR WAS IT A COFFEE MUG? Please check the new first paragraph, especially the end. And maybe move the mathematical fields/Fields Medal diversion to a footnote. Do NOT delete this tag too soon.}} | |
− | + | Field Topology is [https://encyclopediaofmath.org/wiki/Topological_field a subject in mathematics], but in this comic, Randall is instead examining the topology of playing fields used for various recreational activities. The comic strip depicts a situation where the common practice of multi-use athletic facilities has been organized by the "topology department" and constructed to be shared by all sports whose normal playing fields are {{w|topology|topologically equivalent}}. However, one key assumption in topology is that you can ignore the specificities of shape, size and material of the objects concerned. This presents an amusing contrast with the actual activities listed in the comic, where the size and shape of hoops, nets and bars and the material of the field itself can be very significant [citation needed]. | |
− | + | (Not to be confused with {{w|Field (mathematics)|mathematical fields}}, or the {{w|Fields Medal}} prize -- although successfully {{w|Straightedge and compass construction|constructing}} these fields might lead to medals of one kind or another being granted). | |
− | + | In topology, shapes which can be smoothly deformed into one another without adding or removing holes are considered to be "equivalent". Note that a topological hole is an area of the nominal space (or area, or other manifold) through which nothing restricted to this topology can pass. In describing a real-world archway, for example, this would be where the material of the arch is, not the actual 'hole' passing ''through'' the constructed arch, which is the path that one indeed may (or must!) pass through to get from one region of the layout to another. A loop is a path across the allowable territory of a topology (or a viable circuit to make through the world it describes) that end up where it started. If a loop cannot be tightened (ultimately adjusted to take a shorter path) down to a single point, then it must be wrapped around at least one 'topological hole' (i.e. through a physical one), and you have separately unique paths (or points, i.e. on different disconnected topologies) where you cannot adjust one loop to take the route of another, without severing a looped-path and reconnecting it. | |
− | {{w|Baseball}}, {{w|tetherball}} | + | {{w|Baseball}}, and {{w|tetherball}} are played on fields without any holes that the ball or players can completely pass through, so they are ({{w|Group (mathematics)|grouped}}) (physically and mathmatically) into one continuous field without holes. The goals on a {{w|soccer}} field presumably do not create holes because the goalposts and crossbar are connected to the field by the net, so the goals and field are topologically equivalent to a smooth disc. Any path taken into and out of the goal (any number of times) is topologically equivalent to one that does not go into this pocket of space at all. |
− | {{w|Volleyball}} and {{w|badminton}} are played | + | {{w|Volleyball}} and {{w|badminton}} are played on a court through the center of which passes a net suspended from poles, and the {{w|high jump}} has a bar that contestants jump over. The space bounded by the bottom of the net (or bar), the supporting poles, and the ground can be considered to be a hole, a path over and under the net/bar cannot be simplified to one that does not, so their fields all have one "hole". |
− | [ | + | A basketball court has two physical pathable holes, the nets. Parallel bars can be thought of as two rectangles and thus as two topographical "holes". Both have opportunities to path through either (or both) structures, and so the material of the structures define a hole in the topological abstract of the playing 'surface'. Since we are told that these sports fields belong to the Topology Department - and are not necessarily generalized to all sports fields - we may safely assume that their "football" field is for "[https://en.wikipedia.org/wiki/Rugby_sevens Rugby]" which was played at the recent Tokyo Olympics, as American Football uses the Y-shaped goals rather than H-shaped ones. An "H" shaped goal creates a topological hole under the crossbar at both ends of the field, while an American Football field with Y-shaped goals would have no holes. |
− | + | The lane dividers in a swimming pool create bounded holes on the 'playing surface' equivalent to one less than the number of lanes. And each hoop in croquet is a hole with one edge bounded by the playing surface. Similarly, as mentioned in the title text, this configuration is also {{w|homeomorphism|homeomorphic}} to a {{w|foosball}} table (with each rod sustaining the player figures above the table defining a hole) or a {{w|Skee-Ball}} lane (which is even more straightforward, as it is just a plane with several holes in which to throw balls). | |
− | + | Unfortunately, the Topology Department does not seem to have a field for {{w|Hurdling}} events. | |
− | + | ==Transcript== | |
+ | {{incomplete transcript|Do NOT delete this tag too soon.}} | ||
− | + | A row of four signs, each held up by two posts, followed by a row of four rounded lozenge shapes, one for each sign. The signs and lozenge shapes are shaded as if three-dimensional objects, all being flattish with a small third dimension; the four lozenge shapes each have one pair of sides horizontal and the other pair at a slight angle from vertical, denoting a horizontal plane perpendicular to the signs extending "out" towards the viewer, which places each shape "in front" of its sign. All but the first lozenge shape have various numbers of ellipses within the shape - ovoids shaded to denote holes piercing through the objects. | |
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− | + | Leftmost sign: "Baseball. Soccer. Tetherball." The shape below this sign contains no ellipses. | |
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− | + | Second sign from left: "Volleyball. Badminton. High jump." This shape has one large ellipsis in the centre. | |
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− | + | Third sign: "Basketball. Football. Parallel bars." This shape has two large ellipses - one in the top half and one in the bottom half. | |
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− | + | Fourth and rightmost sign: "Olympic swimming. Croquet." This shape has nine small ellipses - eight arranged symmetrically towards the edges of the shape and one in the centre. | |
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− | + | Caption underneath the signs and shapes: No one ever wants to use the topology department's athletic fields. | |
− | :No one ever wants to use the topology department's athletic fields. | ||
{{comic discussion}} | {{comic discussion}} | ||
[[Category:Math]] | [[Category:Math]] | ||
[[Category:Sport]] | [[Category:Sport]] |