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WinterLight
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In colloquial terms, two things are isomorphic if and only if there structurally indistinguishable. For practical purposes, you can’t tell the difference between them.
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greylorn
Guest
Me too, since ballistics tables were not developed for studying. Like trigonometric tables before the days of pocket calculators, they were for looking up values associated with other values. You did not really have to study them, I hope.
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Peggy_in_Burien
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Yes, I really did have to spend time studying ballistics tables; my Senior Master Sergeant made sure I did.
This was in the days before pocket calculators, and even in times of war, we might lose our slide rules! Our ballistic tables had all kinds on information in columns: weight, velocity, altitude, windspeed, wind direction, barometric pressure. All these variables are taken into consideration when determining the exact moment to release a bomb from an aircraft. Bombs, as you know, don’t fall in a vacuum, and, thus, don’t fall straight down. Also, since my job was radar bomb scoring, I needed this information to be able to properly score an “electronic” bomb. The book was as big and as thick as the New York City white pages! And almost as boring.
I was extremely grateful for the day, when the Strategic Air Command gave the duty of studying the ballistic tables over to the Tactical Air Command, in 1980. I closed my book forever, that day, and danced a jig right out in the middle of the scoring room!
Now, tech manuals for radar #9, built in 1939, were huge, but also very fascinating. My radar, the “Pink Pig” was retired in 1992 when SAC stood down for the last time. Picture below. You will also see and hear it at the very beginning of the movie The Right Stuff. Best tracking radar built to that date.
All done by my daughter with computers more than half a world away from the action. I had to be within 25 miles.
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Whadyamean
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Out standing! Not that I ever did it, but RA Heinlein implied that any competent adult had a grasp of ballistics, among other things. Two others of those were GS and epistemology. Those I have dabbled in, but it always fascinated me that if he mentioned a boo, author or whatever, the chances were that you could go to the library and find it. Thanks for sharing your story.
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greylorn
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Peggy,
I stand happily corrected.
Thank you for sharing that fascinating bit of American and personal history. I’m guessing that back when you were flying and mastering tables, I was memorizing B-52 specs on the back of one of my warplane bubble gum cards. Thanks for helping keep me safe enough to do that, and for your service to the United States.When I watch “Right Stuff” for the third or fourth time, I’ll watch for the “Pig.”
I’ve returned to several installations on which I’ve worked, only to find them gone. Kind of sad, but life moves on. Should have taken photos. Have a great year!
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greylorn
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Thank you. If I understand you correctly, then houses in a subdivision built according to the same floor plan are isomorphic, no matter if they are painted differently and occupied by different families.
How does this apply to a mathematical construct that I might understand? For example, are all quadratic equations isomorphic, irrespective of variables?
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Peggy_in_Burien
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The Pig is at the very beginning of the movie, when the narrator is speaking about the attempts to break the sound barrier. It is the van tracking the jet, and that high-pitched “zzzzzzzzzzzihhhhhhh” noise is quite familiar to my ears.Peggy,
I stand happily corrected.
When I watch “Right Stuff” for the third or fourth time, I’ll watch for the “Pig.”
I’ve returned to several installations on which I’ve worked, only to find them gone. Kind of sad, but life moves on. Should have taken photos. Have a great year!
I never flew. I tracked. I also guided aircraft to their drop points. Where did you get warplane bubble gum cards and when? I missed out on those.
One last word on the subject: My Catholic parish school welcomed the arrival of a new principal about 4 years ago. He was introduced to me as a former SAC B-52 pilot. I asked him if he had ever flown over “Courtroom” ( my radar site in West Germany). He said he had. I replied, “I’m Courtroom.” You should have seen his face! He looked mighty surprised to discover one of the airmen who held his career in her hands was barely 5 feet tall, and just as old as he was! I was on crew the day he flew over us on his bombing run. Lots of fun when you run into old airmen.
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WinterLight
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Almost. If they’re painted differently and occupied by different families, then those properties would distinguish the two houses. Two copies of the same floor plan, however, would be isomorphic.
The main idea is that there are no structural differences between two mathematical objects, say two topological spaces or two groups or two graphs. For example, two topological spaces are isomorphic (again, this is a colloquial rather than a techinical explanation) if you turn one into the other and back again. A famous example is a donut and a coffee cup. Imagine, if you will, that both are made of infinitely stretchable material. In that case, you can “deform” the donut into a coffee cup (and back again) without tearing or cutting. (The formal definition is that there’s a continuous bijection, which also has a continuous inverse, between the two spaces.)
Since equations don’t have any obvious structure (they’re really just elements of a particular mathmatical structure, usually, but not limited to, a field or vector space), there’s no structure to compare, and so there’s no obvious way to define an isomorphism between equations.
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