Editing 2625: Field Topology

{{comic
| number    = 2625
| date      = May 27, 2022
| title     = Field Topology
| image     = field_topology.png
| titletext = The combination croquet set/10-lane pool can also be used for some varieties of foosball and Skee-Ball.
}}

==Explanation==
{{incomplete|Created by SOMEBODY HOMEOMORPHIC TO YOUR DOG - Please change this comment when editing this page. Do NOT delete this tag too soon.}}

This comic strip depicts a situation where the common practice of multi-use athletic facilities has been organised and constructed based soley on criteria in which sports are grouped by the {{w|topology|topological equivalence}} of their fields. (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}}, 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 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'. The inclusion of an American football field is perplexing. Commonly, an American football field uses a "Y" shaped upright, making the field topologically equivalent to a plane. However, at lower levels of play (primary and secondary schools), sometimes an "H" shaped upright is used, which creates a topological hole under the crossbar at both ends of the field. The comic might instead refer to Gaelic football or Rugby, both of which use "H" shaped goals and are called "football" in certain contexts. The open-top of the goals is physically an analogue of soccer goals, but as a closed frame of the bottom does not have a net (except in Gaelic Football) these demonstrate topological loops. An alternate explanation would consider passage between the sidebars but at any ''indefinite'' distance above the crossbar (however supported above the ground) to be a special space for scoring purposes, and the topological hole would relate to these limits, held above the ground in a manner not dissimilar to the basketball hoop – although this interpretation is incompatible with 'abstracting away' the special nature of the soccer goal-mouth in its own representation.

The lane dividers in a swimming pool create bounded holes on the 'playing surface' equivalent to 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).

==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 roughly oblong shapes, one for each sign. The signs and oblong shapes are shaded as if three-dimensional objects, all being flattish with a small third dimension. The four oblongs are presented at an oblique angle, as if they are in "front" of the signs extending towards the viewer. All but the first oblong have various numbers of holes "through" them.

zero holes: "Baseball. Soccer. Tetherball."

one hole: "Volleyball. Badminton. High jump."

two holes: Basketball. Football. Parallel bars."

nine holes: "Olympic swimming. Croquet."

Image caption: "No one ever wants to use the topology department's athletic fields."

{{comic discussion}}
[[Category:Math]]
[[Category:Sport]]
@@ Line 8: / Line 8: @@
 ==Explanation==
-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 sports. The comic strip depicts a situation in which 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}}. 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 as the "equivalent" topology department playing fields are actually not very appropriate for the activities listed in the comic, as the standard positioning, size and shape of hoops, nets and bars and the material of the field itself are not equivalent to the real playing fields used for those activities.
+{{incomplete|Created by SOMEBODY HOMEOMORPHIC TO YOUR DOG - Please change this comment when editing this page. Do NOT delete this tag too soon.}}
-(Not to be confused with {{w|Field (mathematics)|mathematical fields}}, or the {{w|Fields Medal}} prize -- although the concept is likely a further pun in the comic, as math (including topology), and most things once can imagine really, are mostly performed ("played") within mathematical fields.)
+This comic strip depicts a situation where the common practice of multi-use athletic facilities has been organised and constructed based soley on criteria in which sports are grouped by the {{w|topology|topological equivalence}} of their fields. (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 equivalent. A topological hole is an area of the nominal space (or area, or other manifold) through which nothing restricted to this topology can pass. 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. For example, when describing the space taken up by a solid object such as a coffee mug, the handle forms a loop with a hole through it. 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", 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.
+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.
-When describing a negative space, such as the space around an archway, the 'hole' would be the material of the arch itself. This is because a loop formed by a ring around any part of the arch material can only be shortened to a finite length, not to a point; the 'hole' is the arch-shaped obstruction which forces the existence of these loops. A {{w|basketball}} hoop connected to the ground forms a similar obstruction with a loop through it, so the space around the hoop contains an equivalent hole. In this comic the topology department has analysed the spaces where various sports are played by the number of such obstructions in the playing area. Each space depicted in the comic is then signposted with the sports which are played on a field with that number of holes.
+{{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|Baseball}}, {{w|tetherball}} and {{w|soccer}} are played on fields which are continuous in three-dimensional space. This means it is possible to traverse any path around or over any of the structures defining the field, while there are no obstructions which can be traversed through in a loop around them. The goals on a soccer field presumably do not create holes because the goalposts and crossbar are connected to the field by the net; Randall apparently considers these to form continuous surfaces which do not allow loops through them.
+{{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".
-{{w|Volleyball}} and {{w|badminton}} are played using a net suspended from poles, and the {{w|high jump}} has a bar that contestants jump over. The structure formed by the net or bar and the supporting poles can be considered to be a "hole" through the playing field, as a path over and under the net/bar forming a loop cannot be contracted to a single point, so their playing fields in the comic 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'. The inclusion of an American football field is perplexing. Commonly, an American football field uses a "Y" shaped upright, making the field topologically equivalent to a plane. However, at lower levels of play (primary and secondary schools), sometimes an "H" shaped upright is used, which creates a topological hole under the crossbar at both ends of the field. The comic might instead refer to Gaelic football or Rugby, both of which use "H" shaped goals and are called "football" in certain contexts. The open-top of the goals is physically an analogue of soccer goals, but as a closed frame of the bottom does not have a net (except in Gaelic Football) these demonstrate topological loops. An alternate explanation would consider passage between the sidebars but at any ''indefinite'' distance above the crossbar (however supported above the ground) to be a special space for scoring purposes, and the topological hole would relate to these limits, held above the ground in a manner not dissimilar to the basketball hoop – although this interpretation is incompatible with 'abstracting away' the special nature of the soccer goal-mouth in its own representation.
-[[File:Double torus illustration.png|thumb|150px|A genus two surface]]
+The lane dividers in a swimming pool create bounded holes on the 'playing surface' equivalent to 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).
-A basketball court has two hoops. {{w|Parallel bars}} can be thought of as two archways. Both have opportunities to pass through either (or both) structures, and so the material of the structures define a hole in the topological abstract of the playing 'space'.  Since we are told that these sports fields belong to the Topology Department - and are not necessarily generalized to all sports fields - we might assume that their "football" field is either for {{w|Rugby_sevens|rugby}} or for American football using H-shaped {{w|Goal (sports)|uprights}}.
+==Transcript==
+{{incomplete transcript|Do NOT delete this tag too soon.}}
-An {{w|Olympic-sized_swimming_pool|Olympic-sized swimming pool}} has ten lanes, and thus nine lane dividers which are fastened to the walls of the pool at each end, creating topological holes through the play area. Each hoop in {{w|croquet}} is similarly a hole through the space; while most versions of croquet use six hoops, nine hoops are used for "backyard croquet" which is played recreationally in the United States and Canada. The fact that the space in a swimming pool is typically filled with water{{citation needed}} has been overlooked by the topology department.
+A row of four signs, each held up by two posts, followed by a row of four roughly oblong shapes, one for each sign. The signs and oblong shapes are shaded as if three-dimensional objects, all being flattish with a small third dimension. The four oblongs are presented at an oblique angle, as if they are in "front" of the signs extending towards the viewer. All but the first oblong have various numbers of holes "through" them.
-As mentioned in the title text, this last 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). These "fields" don't actually have the same number of holes, but are apparently lumped together by the Topology Department as having "many" holes.
-Unfortunately, the Topology Department does not seem to have a field for {{w|hurdling}} events.
-==Transcript==
-:[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.]
-:[Leftmost sign:]
+zero holes: "Baseball. Soccer. Tetherball."
-:Baseball
-:Soccer
-:Tetherball
-:[The shape below this sign contains no ellipses.]
-:[Second sign from left:]
+one hole: "Volleyball. Badminton. High jump."
-:Volleyball
-:Badminton
-:High jump
-:[This shape has one large ellipse in the center.]
-:[Third sign:]
+two holes: Basketball. Football. Parallel bars."
-:Basketball
-:Football
-:Parallel bars
-:[This shape has two large ellipses - one in the top half and one in the bottom half.]
-:[Fourth and rightmost sign:]
+nine holes: "Olympic swimming. Croquet."
-:Olympic swimming
-:Croquet
-:[This shape has nine small ellipses - eight arranged symmetrically towards the edges of the shape and one in the center.]
-:[Caption underneath the signs and shapes:]
+Image caption: "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}}
 [[Category:Math]]
 [[Category:Sport]]