Guest Post: The Mystery of Formica’s Triangles

The following story is by frequent commenter and friend, Tom Peachey. Tom requested that we post his story on our site. We’re more than happy to do so, although we think the story is good enough and funny enough to deserve a more respectable home. Here is Tom’s explanation of how he came to write the story:

My GP has kept up an interest in general science. Some years ago he commented that he was surprised to read that the universe is curved. I resolved to write out an explanation and have finally produced what you see below. It was a no-brainer to use Flatland, but my early attempts were pedestrian. Then I added two characters to voice the two sides of mathematics: one free imagination and the other hard proof.

Marty (and my GP) are encouraging me to publish officially but I can’t think where. So I’m now looking for feedback. In particular, what is the target audience? Please do not send out an electronic version further. I can supply a pdf for printing if anyone would like to try it on their students.


The Mystery of Formica’s Triangles

by Tom Peachey

With ideas borrowed from Edwin Abbott, W. W. Sawyer and Frank Dickens.

copyright © Tom Peachey, November 2020


1. Formica the Flant

Once there was a world who’s name is Forgotten. The highest lifeform there were some ants, but not just any ants. These ants were completely flat. And I mean completely – zero thickness. That was okay because their world was completely flat too. These ants could move forward and back, and side to side, but not up and down – there was no up and down. They could not even imagine up or down.

I forgot to tell you, the ants are called flants in their language, Flantspeak. The picture here shows what they look like.

Despite their lack of thickness, flants are quite intelligent and live civilised and interesting lives. The hero of our story is a flant named Formica. He works as a surveyor. But not just any surveyor, Formica is fifth in line for Surveyor General. Currently he is in charge of a complete remapping of the Queendom. This is his big chance to get promotion to fourth in line.

At home Formica lives with his wife Mica. I should explain, when a male flant marries he takes the name of his wife, just adding the prefix For. They have no children yet, although they apply each year for an egg from the Royal Eggbank, but without success so far with the coin flip.

Mica works as a philosopher, trained in this by her maiden aunts, Saken and Skin. Mica’s work involves sitting and thinking deep thoughts. In Formica’s eyes this is not real work. He is a practical flant, scurrying about the Queendom looking useful. This year he was even chosen to measure the foot of Queen Titude CCCXIV. But Formica pretends to be interested in his wife’s work. That work seems to be thinking about impossible questions, such as why time only runs forward, and why is the universe just two-dimensional. Currently she is worrying whether the universe is infinite or bounded. An infinite universe just goes on forever which seems a waste. But with a finite universe, when you come to the end, there must be something beyond?

Long ago the Queendom had kings instead of queens. But they were brought up spoiled and reigned as gluttonous, cruel and dangerous. The “glorious revolution” replaced kings with queens but they turned out to be gluttonous, cruel and dangerous. Then it was decided to appoint queens at birth. Each would reign until the age of 10 when they would be replaced – before they got too dangerous. But the child queens rarely served a full term, dying from a surfeit of sugar. That’s why there is now an “aleatory monarchy”; all decisions are still made by the queen, but only by answering her advisor’s true/false questions by flipping the royal coin.

Despite having no real power, queens are widely adored, especially the cute ones. For example, the basic unit of length, the foot, is still taken as the length of the Queen’s foot. This is measured each year on the Queen’s birthday, and all recorded measurements are then adjusted to the new standard. This is one reason why Formica is so busy; each year he needs to convert all existing records.

† This name means “attached to Gotten” But no-one can remember what Gotten is.

‡ Not actually flipping, it’s more like spinning.


2. Mica is not Convinced

As we meet Formica he has just completed multiplying all distances in the database by the factor 0.949, and he has time to prepare for the Great Survey. Today he gets home tired and worried.

“You’re worried” says Mica, “what’s ahead?”

“You know I’ll be starting the Great Survey soon …”

“Yes leaving me alone with no husband and no children. Did you know that Tunate next door has had a new baby every year for 12 years running. What are the chances of that?”

“One in 4096” Formica said automatically. “Anyway, Forgerri and I have been testing the new violet laser prtrctrs. They are meant to be incredibly accurate and are very expensive. But it looks like they’re faulty”.

“So take them back to the shop.”

“Not so easy. They are imported from the Kngdm f Md.”

“Md” exclaimed Mica, “they are so backward”.

Yes but …”

“and dirty!”

“Sometimes muddy …”

“They still have kings!”

“But …”

“And they haven’t invented vowels yet!”

“Yes but they are famous for the accuracy of their instruments. That’s why we paid big money for their prtrctrs.”

Mica was starting to see the problem. “What do these prtrctrs actually do?” she asked.

“They measure angles.”

What’s ahead is Flantspeak for what’s up.


“And you want to measure angles because?…”

Formica was happy to talk about his important work. “This Great Survey will make a grid of triangles across the Queendom. We need to measure the triangles accurately.”

“Why?”

“Well, for example – so people will know exactly which foot of land belongs to which farmer. We measure lengths of our starting triangle and for all the other triangles we just measure angles and work out the sides using trigonometry.”

“So … these prtrctrs are not working?”

“They are working, but not accurately. Forgerri and I measured some large triangles, and the angles did not add up to 180 degrees. Always a bit greater.”

“Do you want them to add to 180 degrees?”

“Of course. According to mathematics they must add to 180.”

“Oh mathematics. I missed school the week they spent on that.”

Formica made a smug smile. One week to learn mathematics! He went to Her Majesty’s Special School for Very Smart Flants where they learned all mathematics in a day.

“But” continued Mica, “Aunty Saken did teach me algebra. Show me why the angles must add to 180. What is an angle anyway?”

Formica took out some chalk and drew a line segment. “This is a straight line.”

“What do you mean ‘straight’?”

Mica was in philosophy mode, but Formica was up for the challenge.

“It’s the shortest path between two points.”

“Um …”

“And it’s the path that light takes.”

“Go on.”

“Let’s rotate this line about one end.” He drew the new line.

“This makes an angle – and the measure of the angle tells you how much it has rotated.”

“Okay.”

“If it rotates all the way back to the start, that is a rotation of 360 degrees.”

“Why 360?”

“Well the ancient Babyloniants used 360.”

Mica was about to challenge, but Formica continued quickly.

“And our Queen has decreed it.”

That always won the argument. Mica just nodded. Formica continued.

“So if we just rotate the line half way, we get an angle of 180 degrees on each side.”

Mica nodded again; she was getting to understand this angle stuff.

Formica was now sounding pompous. “I will now prove that the angles in a triangle add to 180 degrees.”

“What is this prove?”

“A proof shows that something must be true, and shows why it is true.”

Mica nodded.

“Now a triangle is made of three straight lines. Making three angles.”

He drew the triangle and coloured the angles.

“I want to show that these angles add to 180 degrees.”

“Extend one side.”

“And at the red corner draw a line parallel to the opposite side.”

He was colouring the new angles to show that they added to 180,

but Mica interrupted. “What is a parallel line?”

Formica was unsure of this but tried to sound confident.

“Parallel lines never meet.”

“How do you know they never meet?”

“Well, you walk along them and check.”

“So if they are parallel, you could walk forever and never decide.”

Formica stroked his antennas for a while, then stomped to his shed and started hammering some nails.


3. A Day Out

The next day was Full Moon Day. No work was done throughout the Queendom. The flants in each village would gather at the local temple to chant prayers. Then each family would walk three times around the temple before adjourning to tend the graves of their ancestors. Mica found this difficult – the other families each had their troop of excited children. She could feel the pity of other flant mothers for her barren family. Or worse. An absence of children was widely believed to be punishment for bad deeds in a previous life. So she was happy when it was time to leave.

Formica was not so happy. The next stop was a feast at his Mother-In-Law’s nest. There, Mica’s mother would quiz him about the chances of grandchildren and discuss at length his shortcomings as a For. But today Formica was not concentrating on his own inadequacies. His mind kept wandering to triangles. Walking home completed a triangle – from home to temple to Mother-In-Law to home. By the time he turned into home he had an idea.

“I have a new proof for the angles of a triangle.” said Formica.

“Does it have parallel lines” replied an amused Mica.

“No. Just walking, and some algebra.”

“Good. I like algebra.”

Formica drew a picture. “This point is H our home.” “Yes.”

“And T is the temple – And M is your mother’s place.”

“Okay.”

“So we have a triangle.” He drew the triangle and coloured the three angles.

“Here T stands for the place of the temple. But I will also use T for the measure of the red angle.”

“And the same for M and H?”

“Yes. I want to show that T + M + H = 180.”

“Now suppose we walk from H to T. Then we turn toward M. We need to turn through an angle of 180 − T degrees.”

Mica inspected the picture. “Yes that looks correct.”

“At M we turn to walk home. This time we turn through an angle of 180 − M degrees.”

“Good again.”

“At home we turn again to face the temple.”

“This time 180 − H degrees.”

“Yes you get it. By this stage we have turned a complete circle, 360 degrees.”

Mica was excited. “Let me solve this.” And she wrote:

“So … they add up to 180.”

Formica reached over and added the standard way to finish a proof.

“Told you so.” he said.

“Let me think” said Mica, “The angles must add to 180. But when you measure them they do not.”

“Correct.”

“It’s as if the straight lines are bent.”

Formica shook his head. “Straight means straight. Bent means bent”.


4. A Flash of Inspiration

At work the next day Formica had this brain itch. Is straight really straight?

“Hey Forgerri! I have a new way to test the prtrctrs. No triangles!”

“How?”

“We put the angles together – like this.” And he drew a picture with three angles.

“Easy” said Forgerri, “no walking needed.”

And they set up a prtrctr and measured three angles like in the picture. Forgerri did the adding up.

“Exactly 180” he said – or rather within 1 second of arc, highly accurate. What is going on? When the angles are together they make 180 degrees, but in a triangle they add to more.”

Formica did not reply, he was transfixed, staring into space. Thinking.

“You should turn off the laser”, said Forgerri.

Formica looked across. “It’s not dangerous. If the beam hits a flant then they get a tickle and move out of the way.” He was amused. “As we speak, the laser might be tickling flants in Md – maybe even further.”

Just then Formica felt someone tickling his tail. He turned, but there was no one there. Then he saw the light. It was violet.

That was when Formica discovered how the universe works. Did he kiss Forgerri and do a little dance? Did he shout “Eureka” and run naked in the town? No, Formica was a serious flant, fifth in line for Surveyor General. He did however recite a little poem remembered from his schoolday when they learnt poetry.

“Then felt I like some Trumpian official
Who starts a war o’er something superficial,
Or, like a flea that meets some other fleas,
Silent, upon a Pekinese.”


 

 

Mica and team measure a huge triangle

 

People have been asking what happened with Mica and Formica. Well, Mica joined the team and went on the Great Survey. While on the road, Mica and Formica developed a new type of trigonometry that works for bent straight lines. It turned out that they were an effective collaboration; Formica would bubble with ideas and Mica would shoot them down, except when she couldn’t. They have become quite famous. The Queen has bestowed a family name – they are now Family Na-Pier. Not that they have time to enjoy their fame. Their life is dominated by baby triplets, three little girl flants: Getful, Give and Eigner. Formica says it must have been a triple-yolker, but Mica disputes this. She claims that someone slipped an extra child into the cot. And she darkly adds “Be careful what you wish for”.

Some readers might be wondering why I have told this story about these flat creatures in this flat world. I’m thinking it might be relevant to a problem in our real 3D universe. As you know, we have at last got messages from our colonies on Alpha Centauri and Lalande 21185. They complete a triangle with our Sun and we now have measurements of the triangle angles. It appears that the angles do not add to 180 degrees, in fact the total is slightly less. Go figure.

Na-Pier is Scots Flantspeak for No Equal.

38 Replies to “Guest Post: The Mystery of Formica’s Triangles”

      1. It’s a reasonable option, but it seems to me Tom’s story would have a much more natural home in some general interest publication. Unfortunately, the editors of such publications tend to sneer at maths people.

        1. Thanks Terry
          I will look into it. All my current work is hampered by lack of access to literature. Libraries no longer provide hard copies and online access is denied to non-staff/students. Universities in Oz have adopted a policy of refusing adjunct status. Both Monash and Queensland have expelled me even though I was publishing papers. The upshot is that I can no longer read the Intelligencer.

    1. Thanks Steve R
      I see that Scientriffic is now part of Double Helix. Must admit I have never read either. Could be a good home for Formica if the audience is (the set of) secondary school students.

      1. I think such school magazines make a lot more sense that the Intelligencer. But it still feels to me the more natural audience is literary and not-very-mathematical adults.

  1. Thanks edderiofer
    I guess I will try to find a copy.
    Followed your link. It seems you are very creative with maths based puzzles.

  2. I see it written that the shortest path between two points is “the path that light takes”. General relativity postulates that a light ray follows a geodesic in spacetime; but whether its path through space (as opposed to spacetime) is a geodesic is another question entirely. I don’t know, and I suspect that answering that question is very difficult, and might depend on details of the spacetime. It presupposes that one can speak meaningfully of a time slice of spacetime in very physical terms. That might seem do-able, but it’s the same as asking “How do we define a/the present moment?”. We can define “now” in flat spacetime–everyone agrees on that–but defining “now” in curved spacetime is not something that the relativity community agrees on. So, unless you can prove otherwise, I think it’s best not to say that light follows the shortest path between two points.

    I might add that the pictures of the ants use the words “top”, “bottom”, “side”, which presumably don’t exist in a two-dimensional world.

  3. You seem to be confusing me with Formica. He learnt classical physics, presumably all in one day, for which light definitely takes the shortest path.
    Again with the pictures, these are the views from outside the flant world, which must be possible because there they are. Actually both Marty and I had the same concern with those images. And in an earlier version I had Formica erasing part of a triangle so Mica could crawl inside to inspect the angles. Another friend is pondering how chalk could write in a 2D world. Yes, one could worry about almost every chapter. I recommend following Aristotle and applying the suspension of disbelief.

    1. In that case, Tom, you should add a caveat for the reader, saying that none of your characters are guaranteed to know what they are talking about, including the one who is presumably doing the teaching. That’s not really the way to get your audience on board, though.

      To say that the flant world has an “outside” is to suggest to the reader that our universe has an outside, since you are, after all, talking about our universe. But the point of the universe’s curvature is that it is “intrinsic”: hence the universe can exhibit curvature without there needing to be an outside.

      1. Don you deserve a fuller reply. Let me start by describing the target audience. It’s the passenger on the Clapham omnibus, or in my case a very smart doctor in a Footscray clinic. The guiding principles of my teaching over the last 60 years are (i) base the teaching on what the recipient knows, or thinks they know, and (ii) make it fun.

        We seem to have the same aim: to expand understanding while avoiding mistakes. Curiously you didn’t mention the more egregious mistake made by Formica when he defined parallel lines by “Parallel lines never meet.” I put that in because I reckon there are a significant proportion of teachers that make this the definition. Rather than jumping in and warning the reader I let Mica tease out the problem.

        With regard to the shortest path, my take on Quantum Electrodynamics is that shortest path is by definition the path that light takes. But if Relativity suggests otherwise then maybe we should wait until I have completed the unification with quantum theory.

        I see that you would prefer me to keep the whole story on an intrinsic footing, which is what Formica struggles with. If you can explain how that would work with my target audience then you must indeed be a great teacher. There is a long tradition of using an embedding in a space of higher dimension to explain an intrinsic geometry, both in research and pedagogy. Popular authors still show a Klein bottle in 3 dimensions even though it is self-intersecting so does not work. I suppose I could have written a worthy tome starting with the axioms of non-Euclidean geometry and deriving the consequences, but I heard a rumour that that had already been done and no copies have yet been seen on Clapham buses.

        1. Thanks for your reply, Tom. Quantum electrodynamics is (almost) always expounded in an inertial frame in flat spacetime. In that case, everything is simple, and the light ray follows a straight line in space. (Of course, we can consider light to be rays here.) Anything else changes the rules, but that is all outside the remit of current QED, as you alluded to. I think that most readers of socratic dialogues assume that everything said by the “all-wise” character is correct, in which case I don’t think it’s a good idea to have that particular character say something that can be questioned. After all, people take little things away from such stories: they might walk away with something you saw as an insignificant detail, which turns out to be all that they remember. Laser beams are real and physical, and making a statement about them is different, I think, to making a statement about the behaviour of parallel lines, since what we mean by “parallel” is a question of definitions. In contrast, a laser beam has an in-your-face reality that transcends how different people might discuss it.

          As for my preferring to keep things intrinsic–no, not necessarily, and in hindsight, I think it’s reasonable to add an extrinsic view, just as you rightly put it when you mentioned Klein bottles. So sure, draw those pictures of the ants. But I do think it’s important to tell anyone who asks the question “What does it mean to say the universe is curved?” that the existence of curvature doesn’t demand extra dimensions for the universe to curve into. Such dimensions might well exist, but they don’t have to. After all, we’re talking about the universe here–a real thing, as opposed to a construction of the human mind like a Klein bottle. I think it’s important to say that physicists who speak of a curved universe are not saying that extra dimensions are a necessary part of their world view.

        2. Hello again Don. You have raised a couple of threads and each time I pull one it just gets longer 🙂

          Looking at the troublesome section on “shortest path” I see that Formica says “It’s the shortest path” and then “And it’s the path light takes”. I guess I was defining the shortest path as that taken by light in order to prepare the reader for the use of the laser instruments, something as you say “real and physical”. Here I write “I guess” because this was unconscious at the time. Many authors report that their characters take over the plot, something I assumed was a literary conceit. But in my case, Mica and Formica would keep me awake at night with their arguments; each morning I would just type it up. There, I’ve shifted the blame to Formica. I am tempted to adjust the section make the light path clearly the working definition of a straight line.

          Socratic dialogue? I must admit that I have never read one. I did try to read the Galileo one, and another by an acquaintance working in History and Philosophy of Science, but both times I was so bored that I gave up after a few lines. So your post sent me to Mother Google to see what a Socratic dialogue is. My story was never meant to have an “all-wise character”. Even the authorial voice of the last chapter is expressing some ignorance – hopefully. The story dialogue is in the older tradition of argument that I imagine the Pythagoreans used to work out Euclid, Book 1? I reckon we can assume that my target audience is at least as ignorant as me.

          Intrinsic versus extrinsic. Thinking as a mathematician, it would be nice to explain how we don’t need those extra dimensions for a curved universe, but I can’t see how to do that within the informal framework of the story. When I put on a physicist hat, I find I don’t care. To me any explanation that uses the existence of something outside our universe, something that has no way of affecting us, is meaningless. Yes my logical positivism is losing popularity. We see the rise of people happy to accept the many-worlds “explanation” of quantum mechanics.

          1. Hi Tom. In defining the shortest path as that taken by light, you are defining your space’s curvature. Let’s think about that. Suppose that you are far from any gravity, in a square-box lift that is accelerating up along the normal to its ceiling due to a rocket beneath its floor. If you put a laser on your left wall and aim it exactly sideways toward the right wall, parallel to the planes of the ceiling and floor, its beam will curve downwards. (It won’t curve downwards if the lift has a constant velocity.) If you now want to define that curve to be a straight line, you cannot avoid describing your right and left walls as skewed relative to each other. But why should the shape of the space be dictated by your choice of where to place your laser? That can’t be, and so the laser’s ray cannot define a straight line in that frame. This accelerating lift is a simple model of gravity, and we only expect things to become more complicated in a real gravity field. And it’s the gravity field that is responsible for the curvature that your story discusses.

            I doubt that Galileo’s dialogue is appealing to most modern readers. Apparently, the real reason he was persecuted by the Church was not from anything scientific he wrote, but rather because he modelled the simpleton in one of his dialogues on the Pope. That’ll do it every time.

          2. Hi Don
            A lovely explanation of the problem of straightness! Even worse, let’s suppose the light beam in your accelerated frame leaves a trace, like in a cloud chamber. What will an accelerated observer see? The light returning to her eyes is also bent which will offset the bendiness of the original? Too hard, my head hurts.

            In mathematics it’s so much easier. We just make a straight line an undefined primitive and let the postulates give it some meaning. And in the naive world of the the target audience, and the pre-enlightened Formica, a straight line can be one of two things. First the path of a stretched string – which is Formica’s “shortest path” and secondly the path of a light ray which is what we use when we look along a piece of timber. Real surveyors use both definitions: a chain or tape measure, and a theodolite. So I guess Formica needs to mention both.

            I was rather chuffed with the walking proof of the triangle sum since it avoids parallel lines. But since it produces a(n) Euclidean geometry, there must be the fifth postulate assumed somewhere in there. Any ideas anyone?

            1. Hi Tom. Sure, what is seen with the eye can be difficult to work out. Many people confuse those two things: the image that is seen with the eye at some given moment, and the reality of where all the molecules are physically at that moment. For example, in “Mr Tompkins in Wonderland”, Gamow made the mistake of basing his visual images on where every object’s molecules were at each moment; that is, he forgot to allow for light-travel time, which is significant in his world where light travels very slowly. He should instead have tackled the very difficult job of figuring out what was seen with the eye. But even well aside from that, the devil is always in the detail in these relativistic discussions.

            2. Hi, Tom. With regard to your walking, what do you get if you try it on the spherical triangle? That should tease out the assumption.

              1. Hi Marty
                For the fifth postulate, let’s use “That, if a straight line falling on two straight lines makes the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which are the angles less than the two right angles.”
                Flantland does not contradict this as every pair of lines intersect. So there must be a contradiction elsewhere. It seems to be that the lines intersect on both sides of the transversal but I cannot see where this contradicts the other assumptions. Thinks: perhaps the 5th postulate is interpreted to mean “… meet \hbox{\em{only}} on that side …” Of course “side” is undefined – I seem to recall that Hilbert remedied this.

                1. Hi, Tom. What I was suggesting was applying your walking proof to the sphere. Hopefully you wont’ end up proving A = B + C = 180 in that case. So, see what breaks down.

                2. Yes I intended to reply to your suggestion but was diverted onto something that seemed more promising!

                  The final diagram in the story, the triangle on a sphere, was programmed using 90 degrees for all 3 angles. There the supplements of the angles add to 270 degrees. So the Euclidean assumption is that angles turned at corners must add to (a multiple of?) 360. Then I started thinking of the equator as the transversal and the other two sides as possibly parallel which led to something similar to the historical discussion – according to my vague memory.

      2. I think that caveat is worthwhile and funny, independent of any valid criticism of the characters. Don, your point that a universe’s curvature is intrinsic is of course fundamental. But I’m not sure that means it’s a bad idea to have an outside to the flant world, and I’m not sure it suggests that our world has an outside. It’s difficult to fit a full course on differential geometry in one short story.

        1. The nice thing about such discussions of curvature is that they can lead into interesting philosophical discussion. Sure, curvature doesn’t demand that the universe have an outside, and yet it’s entirely reasonable to feel that the existence of curvature strongly suggests that there -are- other dimensions that the universe curves into. So, measuring that our universe is curved is a magnificent thing, because it might be the nearest we can get to peering out into dimensions that we aren’t supposed to be able to peer into. But how we might actually observe that curvature directly–rather than observing related quantities–is a difficult question that I’d say is never really addressed by relativity texts. There’s room in that field for novel work.

    1. Hi again Terry
      I seem to recall that you were editor of the Australian Mathematical Society Gazette. Maybe they would consider Formica?

      Mid 70s, I was stuck in my car with a storm raging outside, rather like today. (You were badly hit in Bendigo?) I had just been discussing spoof papers with fellow students, so I spent the time writing up one such. I left it with Derek Holton and went off to Scotland thinking he would submit to the student maths magazine. When I got back, was rather embarrassed to find it in the Gazette – the humour was rather puerile.
      Tom

      1. The same thought crossed my mind (great minds etc.); I have not read the Gazette for a long time; you could ask the editors, but I think a science fiction magazine might be better.

        “The Conversation” ?

        The most famous mathematical spoof article was by Sokal:
        https://physics.nyu.edu/faculty/sokal/transgress_v2/transgress_v2_singlefile.html

        American Mathematical Monthly had an article in 1938 on how to catch a lion which led to this:
        https://scoins.net/bookwork/blog-3/2016-2nd-half/198—how-to-catch-a-lion.html

        And while I am talking about spoofs, when I had finished my PhD I was looking for a job and came across an ad for “A Linear Operator for the Hilbert Space Centre”. Of course it was a joke; it probably started in the tea room, worked its way through to HR, and eventually was published as an ad; I imagine the university paid for the ad – and I suppose that someone got a roasting for it. The PD read well with good use of words such kernel, inverse etc. I should have kept it.

        1. My thoughts:

          1) A sci-fi magazine is still preaching to the converted.

          2) The Conversation is a swamp.

          3) I’m not sure why you brought up spoofs, but Sokal wasn’t a spoof.

          4) Yes, the mathematical theory of big game hunting should be read by everyone.

          5) That linear operator story is absolutely hilarious.

          1. > I’m not sure why you brought up spoofs,
            Terry was reacting to my (tangential) spoof story.

            > but Sokal wasn’t a spoof.
            Please explain.

            1. I think referring to Sokal as a hoax, which is standard, makes a lot more sense. For me, “spoof” announces itself as a joke, and (pretty much) everybody is in on the joke.

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