2625: Field Topology

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Field Topology
The combination croquet set/10-lane pool can also be used for some varieties of foosball and Skee-Ball.
Title text: The combination croquet set/10-lane pool can also be used for some varieties of foosball and Skee-Ball.

Explanation[edit]

Ambox notice.png This explanation may be incomplete or incorrect: Created by SOMEBODY TOPOLOGICALLY EQUIVALENT TO YOU - Please change this comment when editing this page. Do NOT delete this tag too soon.
If you can address this issue, please edit the page! Thanks.

This comic strip depicts a logical extreme of multi-use athletic facilities, in which sports are grouped by the topological equivalence of their fields (not to be confused with mathematical fields, or the Fields Medal prize -- although successfully 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 making or closing cuts or holes, are equivalent. Baseball, soccer, and tetherball are played on fields without any holes that the ball or players can completely pass through, so they are grouped (heh!) into one continuous field without holes. The goals on a soccer field do not create holes; because the goalposts are connected to the field with a net, the goals and field are topologically equivalent to a plane. The same is true of ice hockey, as well.

Volleyball and badminton are played on a field split in two by a net, and the 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, so their fields all have one "hole".

A basketball court has two holes, the nets. Parallel bars can be thought of as two rectangles and thus as two topographical "holes". A football field is a special case. 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 lane dividers in swimming create bounded holes on the 'playing surface' equivalent to the number of lanes minus one. 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 homeomorphic to a foosball table (with each rod sustaining the player figures above the table defining a hole) or a Skee-Ball lane (which is even more straightforward, as it is just a plane with several holes in which to throw balls).

Transcript[edit]

Ambox notice.png This transcript is incomplete. Please help editing it! Thanks.

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."


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Discussion

First 172.70.86.64 12:50, 27 May 2022 (UTC)

Why is football on the two-hole field? Where are the holes? I don't think the goal posts in American football introduce any since they're not closed. Maybe it's soccer? 172.69.68.88 12:58, 27 May 2022 (UTC)

I think it is because the goal posts extend into infinity and the topological definition of a hole: something you can draw a circle around that you cannot contract to a point. [the user placed a horizontal rule instead of a signature by accident.]
Well, you might still be able to call them holes. They would be if they were fully rectangles. --BlackBeret (talk) 12:59, 27 May 2022 (UTC)
Gridiron football's field contains two areas (the endzones) that can be thought of as not being part of the "normal" field of play, for lack of a better way of saying that pre-coffee. Association football likewise has the areas within the nets. Noëlle (talk) 13:05, 27 May 2022 (UTC)
My immediate thoughts were also that football (soccer) and football (gridiron) are the same, or indeed the other way round. In both cases the closed hole (assuming not a Y-like vertical holder, but H-like as per rugby football) plays no more or less topological part. Threading through the hole from behind has no relevence in either, and in fact defining it as a region that is 'a special enclosed gap with meaning' (which doesn't really matter in the topology sense, just like golf would be a topologically hole-less surface and as a coffee-cup's inside 'dimple' doesn't count, just its handle-hole that makes it equivalent to a doughnut) actually counts for something in association football. 172.70.162.155 13:32, 27 May 2022 (UTC)
It's not the space bounded by the goal that is the 'hole' - it's the goal post itself (or in the case of the high jump, it's the bar, not the space under it). The reason soccer doesn't have 'holes' where the goals are is that they're positioned on the edge of the playable area - you can't play around the bars, because as soon as you cross the goal line you're out of play. And it doesn't matter whether it's a Y-shaped or H-shaped goal - topologically, they both form one continuous 'hole'. 172.70.91.80 13:37, 27 May 2022 (UTC)
I don't think that's the reason why soccer doesn't have holes. The goalposts in football are also outside the playable area, and so are the poles in volleyball. I think soccer is listed as zero-holes because soccer goals are typically not fixed to the field, and are instead separate objects that can be dragged around and removed from the field. On the other hand, the same is true of volleyball and badminton nets (and those nets contain many holes!) so the comic seems a bit inconsistent.172.70.175.146 14:05, 27 May 2022 (UTC)
Speaking from a "football is soccer" nation (well, mostly, the exceptional subregions would argue that it's rugby) a soccer goal is typically not draggable around the field, but permanent (or a unit frame that has to be painstakingly hoisted out of the ground if you don't want them in your football stadium, when you repurpose it for other purposes) and it's only the optional net that gets added to the park's permanent goalposts for the official five-aside competition evening or day of the weekend. Draggable goalposts need a further level of intermediate organisation that goes beyond the typical "shipping container with windows cut in it (with shutters bolted over them) as a cheap changing room/officials' cabin" that might be found near the edge of the field but rarely even has as much as a corner flag left in them, between games".
I presume that US 'football' posts are considered holes because they are an infinitely-tall window (even though the delineating poles only reach so high) that is a meaningful slice (where the goal is, you have to loop around it in mutually different unsimplifiable paths to reach the other side), but then that should make for two holes per end, if you count getting a field-goal and then returning round the sides (or vice-versa) as another valid surface-path.
...but, yeah, I can imagine the problem of definition (and cultural famiarity) here is going to produce more problems even than the understanding of topology. One of the less internationally-accepted comics, this. 172.70.85.177 18:51, 27 May 2022 (UTC)
O_O . Randall is united-statesian, so football means the thing where you tackle each other and hold the ball in your hands. I've never been into football, and I've always seen it with two large goal posts with a horizontal bar between them. The hole is formed under the horizontal bar. When I played football in computer games, you had to get the ball over the horizontal bar. After this, I'll search the web to see if the horizontal bar still exists. Regarding soccer, there aren't two holes because the nets are closed at the back. You cannot pass through the field structure by going through a goal: you bump into the net the ball bounces off of when a goal is made. So, Randall is considering soccer fields topologically equivalent to a plane (ignoring all the holes in the netting). 172.70.114.229 14:58, 28 May 2022 (UTC)
I looked up the goal thing and found that what I was imagining are called H-frame or H-style goal posts. Not the norm; the have two posts instead of one. I'm a weirdo that I thought they were what was up. But Randall could have been thinking of H-frame goals. 172.70.230.63 15:04, 28 May 2022 (UTC)
Many high school and amateur football fields still use H-frame goals. The resulting space can be used as a goal in some other sports. That does raise the question of why they didn't just have one field with lots of holes, and just plug the ones up that aren't needed for the sport being played. 172.70.134.191 15:57, 28 May 2022 (UTC)

Tetherball, in many variants, does contain an obstruction -- the pole, which you're not allowed to touch. The Topology Department is getting tired of having to switch out the fields. Noëlle (talk) 13:05, 27 May 2022 (UTC)

But you can surely jump over it, so it's topologically the same as a zero-height pole... 172.70.162.155 13:32, 27 May 2022 (UTC)

Croquet has six hoops and a peg. How does that make for nine holes? Is it including the opponents' two balls as holes? And if so, why aren't opposing players counted as holes in the other sports? 172.70.91.80 13:26, 27 May 2022 (UTC)

[1] 'Nine-wicket croquet, sometimes called "backyard croquet", is played mainly in Canada and the United States, and is the game most recreational players in those countries call simply "croquet".' (Wikipedia) 172.70.126.215 18:58, 27 May 2022 (UTC)

American football goals are Y-shaped. Rugby goals are H-shaped. Did... did Randall get those confused? Also, I fail to see how basketball and American football get two, croquet gets a bunch, but soccer gets zero. Aren't soccer goals (in-game at least) basically the same shape as croquet wickets, just waaaay bigger? Granted, I don't know anything about topology and I came to this wiki specifically cuz I'm dumb, so I'd love if someone could splain this all for me ;) --mezimm 172.69.69.170 13:37, 27 May 2022 (UTC)

The soccer goal has a net, so the ball can't go through it. Topologically it's just a wall (Randall seems to be ignoring all the tiny holes in netting, presumaby because they're smaller than the balls so they're insignificant to the sports). Barmar (talk) 14:10, 27 May 2022 (UTC)
I agree with that explanation - the net is the only thing that makes the soccer field not to have holes. It should be included in the comic explanation.
The hole for the volleyball only makes sense taking in account that the bottom of the net doesn't reach the floor, although this space is not used in the game.--Pere prlpz (talk) 14:18, 27 May 2022 (UTC)
I agree about soccer; the explanation should be that soccer goals (with net) are topologically part of the plane. The same is true of ice hockey, even though you can travel "around" the net, it is topologically part of the field with no holes. As for (American) football, the topology only makes sense for H-shaped goals, which are more often seen on primary/secondary play fields than in higher level play. Aramisuvla (talk) 16:03, 27 May 2022 (UTC)
Agreed. Soccer goals are shaped such that their bottoms connect smoothly to the ground in a single continuous piece. So they are topologically equivalent to the plane. This wouldn't be the case if not for the back part holding the net. That's unlike basketball hoops, which are actual holes. The holes in football must be referring to the H-shaped uprights that were standard until 1967 in professional leagues and are still seen in some high school fields and even a couple college fields. 172.70.131.128 03:08, 28 May 2022 (UTC)
EDIT: I should point out that the net actually has, like, hundreds of holes. But I think the net here is being treated as a continuous sheet. 172.70.126.215 03:10, 28 May 2022 (UTC)
I mentioned all the little holes in the net in my comment that you're replying to. Barmar (talk) 16:43, 28 May 2022 (UTC)

The group link pointing to group (mathematics) doesn't bear any relation with the sentence or the comic. I would remove the link.--Pere prlpz (talk) 14:18, 27 May 2022 (UTC)

The joke seems important to me because their no consideration of the word 'field' being a math pun, and it raises the idea in readers. 162.158.79.74 15:11, 28 May 2022 (UTC)
When I first saw the comic title I assumed that part of the joke would be a pun on the word "field" being used for both sports and math. And even though the comic doesn't explicitly make this joke, I'll bet it inspired Randall. It's worth mentioning. Barmar (talk) 16:43, 28 May 2022 (UTC)

In rugby (both League and Union) the goalposts are within the field of play: significant game activity takes place behind them. This is not the case with soccer. I have no clue what difference this makes topologically. [[Special:Contributions/|]] ([[User talk:|talk]]) (please sign your comments with ~~~~)

I've been thinking about this classification system, and can't quite work out the baseline for it. I think we're supposed to assume that the whole 3d manifold is represented in a 2d 'field', or at least any path through the air flattened to an arbitrarily thin surface 'bulge' during topological rationalisation. But there are several possible field-of-play definitions we can be using...

  • A single valid 'play' or traversal
    • For ball-sports (or indeed other play-objects) this could be where the item can travel. But in this case I think almost 'all' codes of football are Type 1 (first of the topologies) as almost every football code deals with both 'goal' and 'endzone' (where valid) as the same as a hole (dimple) in golf... It goes into it and it might as well come out of it again, there's no continuation of play 'through the defined' space, and so the topological hole (the barrier defined the scoring membrane's edge) never comes into play.
      Unlike in Gridiron, where a touchdown doesn't even need the 'ball' to touch the ground, rugby (league and/or union, and possibly further derivatives) requires this and a player can fail to score a Try if (s)he passes bodily over the line but is unable to plant the ball (not allowed to throw/drop it) and I'd have to check what happens if the defending player(s) keeping them sufficiently off the ground (assuming that's done in an allowable fashion) returns the intended scorer back over the line via a circuitous route around /back-through the suspended goal-mouth (above the cross-bar, between the verticals)... They keep changing those kinds of technical rules, so I can't be sure of the current technicalities involved.
      Likewise, a volleyball or shuttlecock that passes under the net-top-edge is out of play, so it is really a Type 1 under this definition. (Might as well be a solid barrier, floor-to-top-height, rather than a thin bar or a partial net.)
      The basketball case is interesting. Although a dunk ends the play of the ball, I'm not sure if the path of a ball up through the hoop does not. In that circumstance I could believe it is a Type 3 case, but if that's a game-stopping thing then Type 1.
      Croquet is indeed a varying number of paths through (I think) an unordered set of holes, or at least nothing to say that they can be taken out of order (or 'un-passed-through'), and you can't necessarily restrict a 'play' to one shot at a time if certain conditions allow you to play on, so dodging in and around all scoring zones defined by the hoops gives you something like.
    • For player/competitor/participant movement, similarly passing under the bar is not valid for the High Jump.
      I don't think there's anything to stop such transitions upon the Parallel Bars, but it is much more a feature of the Uneven Parallel Bars, whereas from what I've seen of the sport, the even-variety tends to be topologically used much as the pommel-horse.
      Players of football (American variations certainly, rugby of course, proper football if you don't bother with the nets) are not restricted from passing through the scoring area (either way) on a circuitous path that may be off the field of play but isn't off the field of players.
      For the Olympic Swimming, I'm not suring porpoising over and under the lane-delineations is a thing, so I would have said that (under this definition), it should be a number of entirely disconnected Type 1 'zones', with no valid movement between them at all.
      So far as I'm aware, there are no rules for/against croquet players passing through hoops (intentionally or perhaps because they severely annoyed an opponent) so maybe that stands in this case, too. Ditto for basketball, if hoisted. Although in both cases it may prevent the balls passing through immediately afterwards, without game-stoppage to resolve the issue.
  • If it's a game's-worth of play, then the status of the basket in basketball (unlike the pocket in snooker/pool/some-versions-of-billards) might be defined by the topological-hole-that-is-the-physical-hole's-edge, rather than treat it as the old basket-with-bottom from which the precusor to the net-ring almost immediately evolved. And the same could be said about the suspended scoring-hole (whether supported as Y-post or an H-post, the lower limb(s) are merely physical necessities that play little part in the gameplay specifics except as a general hazard to avoid, it is the crossbar and verticals-to-infinity (and the infinity itself) that is the gap through which a circular path cannot be rationalised back to a point). For most of the rest (including the participant-paths, with there being nothing to stop the traversal of a footballer of whatever stripe jumping the cross-bar, but that may only mean something in the topology of some variations, as far as the game is concerned...) it seems meaningless. Even in an Aussie Rules field with four 'posts' per end, and probably more interest in whether jumping onto an opposing player is against the rules or indeed an entirely legitimate and expected tactic.
  • The general arena-wide area is a further superset (perhaps with no additional complications, i.e. exactly congruent) of the field-of-play(er) definition. For coin-operated table-top games (foosball/table-football) the path from each goal may (additionally to any on-top topological loop-disconnections) force passage of the ball underneath and out into the new-play insertion spot. So add a couple more (unidirectional) paths, at least. Or six for a coin-operated pool/etc table, and I assume the Skeeball (not something I'm familar with, at least by that name) is defined that way already...

Sorry, I found I needed to say a lot more than I thought I did, so the first point (and sub-points) went on a bit and I cut down what I might have said for the following points. I may come back to re-edit this. I've got a handy little table, in mind, but I'm not sure it'll work much better to summarise everything I've been cogitating about for most of today while away from the keyboard... 172.70.162.5 15:57, 28 May 2022 (UTC)

The "hole" in the goalpost in American football is relevant for field goals, not touchdowns. Barmar (talk) 16:43, 28 May 2022 (UTC)

As has been alluded to, this must be an American university's topology department. A rest-of-the-world university would include four holes for cricket. 172.70.134.191 17:48, 28 May 2022 (UTC)

Ok, this is my (not yet properly tabularised, or properly wikimedialinked) idea of all the kinds of information I'd suggest go in there.
But it's a monstrocity and I don't want to remove the very useful existing information already in the Explanation (that may even be better/more accurate than my interpretation).
...so here it is for review. If anything in it is useful to anybody else as inspiration for future edits then... well, your choice!

  • Click to expand:
Competition
|-
Field diagram
|-
Usage description
|-
Topology
|-

Type 1 Field
|-
(First image in comic.)
|-
Any path looping around this area can be moved at will and shrunk to just one point that could result from any other path.
|-
A homogonously flat lozange surface with no other notable features.

Baseball
|-
(Partial!) https://en.wikipedia.org/wiki/Baseball#/media/File:Baseball_diamond.svg
|-
The playing area for baseball contains many important physical features for scoring and playing purposes, but is essentially one flat area (and continuous airspace) when you disregard the elevation of the pitcher's mound or even the outfield fence  and stands (for any ball that carries that far, upon being hit).
|-
Randall explicitly classes this in the Type 1 diagram, and there isn't any obvious reason to argue this point.

Association Football ("Soccer'"/"Football")
|-
https://en.wikipedia.org/wiki/File:Football_pitch_metric_and_imperial.svg
|-
An unobstruted rectangular playing area with a goal formed of two vertical posts connected between the tops by a crossbar. In official competition (and where otherwise desired) there is a net stretched behind each goalmouth to stop any ball that passes completely through it (with or without hitting any of the posts), although games can be played with no net in place, or in street/schoolyard situations by goals defined only as a goalpost-like markings painted upon a solid wall (hitting the  wall within the bounds of the painted line constitutes a goal, give or take arguments about whether it counts if it hit the line).
|-
Stated by Randall as a Type 1 (a single unobstructed zone), which is likely due to the 'pocket' of the net-backed goalmouth being nothing more than a straight extension of the playing area.
However, an un-netted set of goalposts might be considered a Type 3, with each set of goalposts defining an impassible frame (the hole in the topology, not the same thing as the physical hole formed by the goal-frame) within which the balls can freely pass and return not through the goalmouth, or vice-versa.

Tetherball
|-
https://en.wikipedia.org/wiki/File:Tetherball_in_Georgetown,_Seattle,_Washington.jpg
|-
A ball attached to a cord anchored at the tip of a pole that is in turn stuck in the ground.
|-
Although the mechanism used to allow free swivelling of the tether around the pole may be quite complex (including being looped around a helical thread to help register how many excess orbits of the pole the ball has made in either direction), the basic premise can be simplified to a single extrusion from the playing area, which is topologically identical to a playing area with no extrusion at all. Thus Randall properly states this as a Type 1 variant.
 
Type 2 Field
|-
(Second image in comic.)
|-
Any path that canot be shrunk to just one point will be pass around the unpassable hole in the topology.
|-
A homogonously flat lozange surface with a single central hole in it.

Volleyball
|-
https://en.wikipedia.org/wiki/Volleyball#/media/File:VolleyballCourt.svg
|-
A volleyball court consists of a flat area disected by a raised net in the centre. Valid shots pass over the net, but it is possible for the ball (or players) to pass between the net and the floor.
|-
Randall lists this under the Type 2 diagram. An argument can be made that the net could effectively reach to the ground, or questions asked about anchoring the net top/bottom to the posts at either side with separate straps (adding left and right 'passages' between the elements of the obstacle that is the net) but he clearly intends the loop around the hole to represent the ability to passing over the net one way and under the net the other (or vice-versa) as a topologically irreducible loop.

Badminton
|-
https://en.wikipedia.org/wiki/File:Badminton_court_3d.svg
(Note that this diagram completely abstracts the under-net area away.)
|-
The net setup is very similar to volleyball, i.e. raised above the ground, with very similar rules regarding valid shots between the areas on each side.
|-
As with Volleyball, Randall feels justified in this being classed as a Class 2, having similar reasons for this as well as possible arguments against.
(Note that another form of Badminton is arguably far more topologically complex!)

High Jump
|-
https://en.wikipedia.org/wiki/File:1912_Platt_Adams5.JPG
|-
A bar supported at height between two supports. The idea is to successfully pass over the bar (without knocking it off, the bar being only supported to the supports, not firmly attached to them), although a competitor who decides to abort their attempt mid-run might well choose to pass underneath to default the attempt with the least physical and organisational aftermath.
|-
With an 'above' and 'below' path to potentially loop around (though not in a single jump), Randall chooses to ascribe this as a Type 2. If a competitor displaces the bar, during a failed jump, it can morph the topology into a Type 1 scenario

Type 3 Field
|-
(Third image in comic.)
|-
Any path that canot be shrunk to just one point will pass around one ot other or both of the holes in the topology.
|-
A homogonously flat lozange surface with two holes in it, towards each end.

Basketball
|-
https://en.wikipedia.org/wiki/File:Basketball_terms.png
|-
Played upon a court, at each end of which is a tall pole (or supporting wall or other structure) from which a 'basket' is projected over the playing area. The earliest baskets were an actual closed-bottom basket, but this required climbing up to retrieve balls successfully landed within them. By removing the bottoms of the baskets and, later, using just a hoop (with or without a bottomless net). Points are scored by sending the ball through the basket-loop from above, to be retrieved for further play as it exits below.
|-
Topologically, the edge of each loop is directly connected to the ground, so it can be smplified as a two-hole Type 3 field (the hole in the field is the impassible rim in the basket-loop). This does not preserve the orientation (or intended unidirectional nature) of the basketball-shot, but this is Topology's fault, not Randall's!

American/Canadian Football ("Gridiron"/"Football")
|-
https://en.wikipedia.org/wiki/American_football_field#/media/File:AmFBfield.svg
https://en.wikipedia.org/wiki/File:Canadian_football_field.png
|-
A unobstructed rectangular playing area and two 'Endzones' at each end. Goalposts are either of an "H" shape or essentially a "Y" (crossbar, upper verticals and a single utilitarian post, usually set back beyond normal playing area with an extension over to hold the crossbar directly over the goal-line. The verticals are tall but are also conceptually projected upwards without limit, for scoring purposes, should a field-goal/etc be kicked high enough to exceed the structures.
|-
Stated by Randall as a Type 3 (a topological hole at each end of the field), which may represent the bound surrounding the elevated goal-scoring area. Alternately it represents the physical structure of the H-shaped posts which rationalise down to the open-backed ground-touching goalpost footings and the crossbar.

Parallel Bars or perhaps Uneven Parallel Bars
|-
PB: https://en.wikipedia.org/wiki/File:AlejandroonParallelBars.jpg
UPB: https://en.wikipedia.org/wiki/File:Paksaltoliukin.jpg
|-
The Parallel Bars are two horizontal bars supported at roughly hand-height, upon which a gymnast will perform various hand-supported feats strength and coordination. The participant will not usually fully use the space beneath either bar (and between the two supports for the bar), but a  will needs the opportunity to grip fully around the bar, especially when the other hand is released for a complicated body movement and it would be impractical or a different discipline entirely to used a 'filled' bar-support.

The Uneven Parallel Bars are two similarly supported bars but at two different (and greater) heights, with the performance being generally that of keeping the grip of both hands (or knees/etc) on either one or other of the bars whilst rotating around its axis, when not actively transfering across between the bars themselves.
|-
Effectively two loops (as per basketball hoops but in a different orientation and scale). The Type 3 topology suggested by Randall is more meaningful for the use of Uneven Parallel Bars, but is probably applicable to the 'even' version in its own way.

Type 4 Field
|-
(Fourth image in comic.)
|-
Any path that canot be shrunk to just one point will pass around at least one (and possibly several) of the nine holes in this topology.
|-
A homogonously flat lozange surface with nine small holes dotted into it.

Olympic Swimming
|-
https://en.wikipedia.org/wiki/File:Swimming_pool_50m_2008.svg
|-
In competitive swimming, a swimming pool is often delineated into lanes (for Olympic purposes, Lane 0 to Lane 9, though usually not all will be used) by floating barriers and other markings. These provide a limited amount of wave-reduction but mostly keep competitors from inadvertently drifting across or into each others' paths.
|-
Randall considers this setup to require nine 'holes' in the competition area, presumably where the floats pass along the surface of the water, to make a Type 4 field of competition. He must then consider it perfectly possible for competitors to pass under or over these barriers, at will, with complete disregard for the usual competition (and risking disqualification). Otherwise, it might be best considered as (up to) ten separate Type 1 arenas, with just one swimmer in each.

Croquet
|-
https://en.wikipedia.org/wiki/File:Modern_croquet_equipment.JPG
|-
A game in which a number of metal hoops are placed in the ground such that a given number of players (or teams of players) must each propel their own ball(s), and possibly those of their opponents, through each loop either directly with their own mallet or through contact between balls.
Many variations exist with differing numbers of hoops and variations of rules and winning conditions. Randall appears to favour the "Nine-wicket Croquet" popular to North America.
|-
The topological simplification of nine hoops across a flat surface can be thought of as the Type 4 topology displayed.

Table Football ("Foosball"/"Table Soccer") - as per title-text
|-
https://en.wikipedia.org/wiki/File:Foosball_garlando_aerial.jpg
|-
An enclosed playing surface with (typically) eight rotatable and extendable bars supporting representative (soccer) 'footballer' figures, ready to strike a small ball across the surface, as might be desired by the two or more opposing players who are each able to control the movements of half of the 'bars' (each team's-worth having a goalkeeper, defence, midfield and attacking 'layer'). By skill and/or luck, the aim is to propel the ball into the opposing's player's goal.
On coin-operated games, often the playing area is usually sealed off from direct manual interference, and a ball that goes into the goalmouth finds itself in a lower chamber that stores the ball(s) and deposits them via some feed to carry the ball back up and 'thrown in' towards the centre of the table to start the next attempt at goal.
|-
With eight bars across, and potentially two goalmouth sinks, this may not actually add up to a nine-hole Type 4 field of play. But presumably Randall is thinking of a version that does.

Skee-Ball
|-
https://en.wikipedia.org/wiki/File:Skee_Ball.JPG
|-
An arcade game in which a ball is propelled by the player to land in (according to skill) one of various holes in a target-ridden surface (to return back to the player for another go).
|-
It would depend upon the exact confuguration of Skee-Ball machine but, again, Randall seems to think this matches the Type 4 topology.

Further (football) examples, unmentioned
Australian Rules Football ('Aussie Rules'
|-
https://en.wikipedia.org/wiki/File:Footygroundfix.svg
|-
An unobstructed oval field with four simple vertical posts upon the perimiter arcs at each end.
The ball passing between the (taller) central pair of each end's posts (projected upwards indefinitely) is a Goal. Passing between the outer posts and the adjacent central one (or bouncing off these) is a Behind.
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Type 1 if the protruding poles are rationalised to zero, without respect to scoring zones. Four or perhaps six topological holes (two or three per end) if respecting the imaginary projections indefinitely upwards for scoring purposes, depending upon if you care about chirality of the ball path.


Gaelic football ('Gaelic') - fields also used for Hurling
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https://en.wikipedia.org/wiki/Gaelic_football#/media/File:Gaelic_football_pitch_diagram.svg
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An unobstructed rectangular field with an H-shaped set of goalposts at each end, the area below the crossbar often being netted, while the upper verticals being nominally considered as projecting upwards without limit.
Valid balls sent over the crossbar and between the verticals are awarded Points; those sent into the netted goalmouth are Goals (equivalent to three Points for scoring purposes).
There is no in-play use of the area behind the line of the goalposts, unlike various other football codes with similar-looking posts.
|- Topologically, probably considered a Type 1. Goal-shots are into a 'pocket' extension (if nets are used), and Point-shots are topologically indistinguishable from passing over any other part of the boundary line.


 Rugby League/Union ('Rugby'/'Rugby Football'/'Football')
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https://en.wikipedia.org/wiki/Rugby_union#/media/File:RugbyPitchMetricDetailed.svg
https://en.wikipedia.org/wiki/Rugby_league_playing_field#/media/File:NRL_Rugby_League_field.svg
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An unobstructed rectangular playing area and two 'In Goal' areas continuing on behind the 'Try Line' upon which the H-shaped goalposts sit.
The field of play extends into this area, the lower parts of the vertical posts play no purpose other than to hold the upper elements in the air. A 'Try' (roughly equivalent to a Touchdown) can be scored by placing the ball somewhere over the line or by touching the base of the (often padded) posts.
The cross-bar and the verticals upwards of it (towards and bounded at infinity) count as the hard boundary of a scoring area for "conversions" (taken immediately after a try) and other kicks (penalties and drop-goals).
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Might be treated as Type 3 (two holes), unless concerned about whether balls kicked through the goals or taking across the try line weave back one or other side of, or between, the lower vertical posts.
Alternately, is a Type 3 for the lower (not more special for scoring than any adjacent lower area) frames, while the open tops (meaningful for scoring purposes) rationalise as topologically irrelevent.

(TL;DR; - It's too long, you may not want to read it...) 162.158.34.213 21:47, 28 May 2022 (UTC)