Collections of mathematical objects
February 16, 2022 5:53 AM Subscribe
If math is the art of finding relations between abstract objects, then a
catalogue of abstract objects is a good place for a mathematician to start. So: real numbers (in order of popularity),
equations,
functions,
formulas involving π,
tilings (previously, specifically nonperiodic),
rings and their properties,
finite group representations,
packings of equal circles in a square,
triangle centers (previously),
top ten lists of prime numbers,
integer sequences (previously, extremely previously),
combinatorial statistics,
graphs,
movies,
fundamental theorems,
counterexamples.
I wish there were some criteria for what makes a prime number of a certain type a "Top 10"..It seems like sometimes it's size, but not always right? They're just the curator's favourites? But what does the curator look for in a prime number? Is it the same for all types of primes or does it vary?
posted by If only I had a penguin... at 7:07 AM on February 16, 2022
posted by If only I had a penguin... at 7:07 AM on February 16, 2022
collection of ways of ranking prime numbers
posted by polytope subirb enby-of-piano-dice at 7:11 AM on February 16, 2022
posted by polytope subirb enby-of-piano-dice at 7:11 AM on February 16, 2022
Happy to have an extremely previously from 2002, almost 20 years ago.
The first link is really interesting. I can see usefulness of this. If your iterative algorithm keeps spitting out some real number and you wonder whether it might have a closed form, this might help point you in the right direction.
posted by vacapinta at 7:13 AM on February 16, 2022 [2 favorites]
The first link is really interesting. I can see usefulness of this. If your iterative algorithm keeps spitting out some real number and you wonder whether it might have a closed form, this might help point you in the right direction.
posted by vacapinta at 7:13 AM on February 16, 2022 [2 favorites]
I wish there were some criteria for what makes a prime number of a certain type a "Top 10"
Obviously the best prime number is 2^6972593 − 1 because it's not too big and not too small, and all you need is a light supercomputer.
posted by AlSweigart at 7:41 AM on February 16, 2022 [3 favorites]
Obviously the best prime number is 2^6972593 − 1 because it's not too big and not too small, and all you need is a light supercomputer.
posted by AlSweigart at 7:41 AM on February 16, 2022 [3 favorites]
a light supercomputer.
To store 871,574 bytes of 'ff' in memory probably doesn't need a supercomputer. Admittedly 640K won't quite fit that number, but still.
posted by polytope subirb enby-of-piano-dice at 7:48 AM on February 16, 2022
To store 871,574 bytes of 'ff' in memory probably doesn't need a supercomputer. Admittedly 640K won't quite fit that number, but still.
posted by polytope subirb enby-of-piano-dice at 7:48 AM on February 16, 2022
primes that resemble flies from a distance
posted by flabdablet at 8:18 AM on February 16, 2022 [2 favorites]
posted by flabdablet at 8:18 AM on February 16, 2022 [2 favorites]
To store 871,574 bytes of 'ff' in memory probably doesn't need a supercomputer. Admittedly 640K won't quite fit that number, but still.
Storing it is easy, checking it is hard.
posted by mhoye at 8:31 AM on February 16, 2022 [2 favorites]
Storing it is easy, checking it is hard.
posted by mhoye at 8:31 AM on February 16, 2022 [2 favorites]
Large numbers are a land of contrasts.
posted by polytope subirb enby-of-piano-dice at 8:44 AM on February 16, 2022
posted by polytope subirb enby-of-piano-dice at 8:44 AM on February 16, 2022
How to recognize prime numbers from a long way away...
posted by Billiken at 9:25 AM on February 16, 2022 [1 favorite]
posted by Billiken at 9:25 AM on February 16, 2022 [1 favorite]
Run Length Encoding could easily fit that into 640K, it's only a handful of bytes.
posted by zengargoyle at 11:43 AM on February 16, 2022 [1 favorite]
posted by zengargoyle at 11:43 AM on February 16, 2022 [1 favorite]
That's a good point, you could write it something like "2^6972593 − 1".
posted by polytope subirb enby-of-piano-dice at 12:02 PM on February 16, 2022 [1 favorite]
posted by polytope subirb enby-of-piano-dice at 12:02 PM on February 16, 2022 [1 favorite]
Meh, 0x1101a992cd will get you there in 5 bytes.
posted by zengargoyle at 12:50 PM on February 16, 2022
posted by zengargoyle at 12:50 PM on February 16, 2022
Oops, make that 0x1106a64b0d. Sorry.
posted by zengargoyle at 12:56 PM on February 16, 2022
posted by zengargoyle at 12:56 PM on February 16, 2022
Damn, make that 0x8406a64b0d.
First byte: first bit is 1/0 the rest of the 7 bits are how many bytes to read.
So in this case that turns out to be 0b1000.0100 or a single 0b1 for the 1/0 and a 0b0000100 for the second. Read the next 4 bytes.
Given 871_574 'ff' as the answer, 0xff is 8 bits so 871_574 * 8 is 6972592 or 0x06a64b0d which is the 4 bytes with padding to not go all nibbly or bitly.
Still 5 bytes so a handful.
posted by zengargoyle at 1:17 PM on February 16, 2022
First byte: first bit is 1/0 the rest of the 7 bits are how many bytes to read.
So in this case that turns out to be 0b1000.0100 or a single 0b1 for the 1/0 and a 0b0000100 for the second. Read the next 4 bytes.
Given 871_574 'ff' as the answer, 0xff is 8 bits so 871_574 * 8 is 6972592 or 0x06a64b0d which is the 4 bytes with padding to not go all nibbly or bitly.
Still 5 bytes so a handful.
posted by zengargoyle at 1:17 PM on February 16, 2022
Here are a few more:
posted by mhum at 1:41 PM on February 16, 2022 [2 favorites]
- The Stony Brook Algorithm Respository
- The Complexity Zoo and The Complexity Garden
- Catalog of Small Graphs
- Brian McKay's collection (a bunch of small graphs of different types and some Latin Squares as well as a list of Hadamard matrices up to order 32))
- Ian Wanless' collection (lots of Latin Squares stuff)
- Ted Spence's collection (some graph, design, and Hadamard matrix stuff along with a complete characterization of Steiner systems S(2,4,25))
- List of matroids up to rank 9 (just in case you needed a matroid of rank less than or equal to 9)
posted by mhum at 1:41 PM on February 16, 2022 [2 favorites]
Somebody please explain "rings" (and where the name came from) plz. I sort of get that integers are not a ring and real numbers are but what's the point? Is there like a linking rings trick where real and oh say surreal numbers get joined? (yes there are surreal numbers) (which I really don't get either).
posted by sammyo at 5:09 PM on February 16, 2022
posted by sammyo at 5:09 PM on February 16, 2022
Ok I did go to the "ring database", first entry: C[0,1] the ring of continuous real-valued functions on the unit interval, so not helpful, ok, I'll be back in a few years after enrolling in a few analysis and topology courses. Sigh.
posted by sammyo at 5:14 PM on February 16, 2022
posted by sammyo at 5:14 PM on February 16, 2022
You don't need analysis, topology, or anything more advanced than group theory to understand what a ring is. I'm going to be going for slightly broad strokes here because it's 3am.
A group is a set (finite or infinite) of objects and a rule (operator) for combining any of them. The operator has to satisfy a few additional conditions: one object is an identity and doesn't change the other object it's combined with, each object has an inverse that combines to make the identity, and the operator is associative so you can apply it to 3 objects in any order.
Examples of a group are the integers with addition, the nonzero real numbers with multiplication, or the odd and even integers with addition.
A semigroup is like a group, but even less restricted. It just needs associativity. Examples of a semigroup are the integers with multiplication. Or the integers greater than 100 with multiplication.
A ring is a set of objects with two operators. One of them (sort-of-addition) has to form a group on the set, the other (sort-of-multiplication) has to form a semigroup on the set, and they need one more condition: distributivity. That's the identity a*(b+c) = a*b + a*c which you probably recognise from algebra classes in school.
Examples of rings are the integers with addition and multiplication, NxN matrices with addition and matrix multiplication, and complex numbers with addition and multiplication.
(The thing that integers aren't and real numbers are is a field, which like a ring except that the multiplication operator needs to form a group instead of a semigroup when the addition-like operator's identity is removed)
You can do a lot of mathematics just from those definitions; the basic theory of groups and rings is very rich, even before getting into topics like algebraic topology (which I absolutely have not gotten into).
posted by polytope subirb enby-of-piano-dice at 7:29 PM on February 16, 2022 [2 favorites]
A group is a set (finite or infinite) of objects and a rule (operator) for combining any of them. The operator has to satisfy a few additional conditions: one object is an identity and doesn't change the other object it's combined with, each object has an inverse that combines to make the identity, and the operator is associative so you can apply it to 3 objects in any order.
Examples of a group are the integers with addition, the nonzero real numbers with multiplication, or the odd and even integers with addition.
A semigroup is like a group, but even less restricted. It just needs associativity. Examples of a semigroup are the integers with multiplication. Or the integers greater than 100 with multiplication.
A ring is a set of objects with two operators. One of them (sort-of-addition) has to form a group on the set, the other (sort-of-multiplication) has to form a semigroup on the set, and they need one more condition: distributivity. That's the identity a*(b+c) = a*b + a*c which you probably recognise from algebra classes in school.
Examples of rings are the integers with addition and multiplication, NxN matrices with addition and matrix multiplication, and complex numbers with addition and multiplication.
(The thing that integers aren't and real numbers are is a field, which like a ring except that the multiplication operator needs to form a group instead of a semigroup when the addition-like operator's identity is removed)
You can do a lot of mathematics just from those definitions; the basic theory of groups and rings is very rich, even before getting into topics like algebraic topology (which I absolutely have not gotten into).
posted by polytope subirb enby-of-piano-dice at 7:29 PM on February 16, 2022 [2 favorites]
Yeah, rings and fields tend to end up like another "let's explain monads" sort of thing. But like polytope subirb enby-of-piano-dice opines... not really hard, you probably already know it. If you take 0 to Infinity as Integers you can do addition, any Integer plus any Integer is still another Integer. And you can multiply, any Integer times any Integer is an Integer. The set of things and the operations you can do on them always lands in the same set you started with. That's a thing. You can't do division like that 4/2 might be 2, but 5/2 is 2.5, you've escaped the set you can't do that. That's another sort of thing, not an integer.
You do this every day by looking at the clock. 0-11 mod 12. 11+2=1, 6+12=6, all the set of objects and the operations end up in the same set. That's a thing.
I can't keep things like groups/rings/fields in precise maths definitions but they're not hard to grock and use.
But yeah, a little bit of group theory. Details are maths pedantic a bit about categorizing the details.
posted by zengargoyle at 8:58 PM on February 16, 2022 [2 favorites]
You do this every day by looking at the clock. 0-11 mod 12. 11+2=1, 6+12=6, all the set of objects and the operations end up in the same set. That's a thing.
I can't keep things like groups/rings/fields in precise maths definitions but they're not hard to grock and use.
But yeah, a little bit of group theory. Details are maths pedantic a bit about categorizing the details.
posted by zengargoyle at 8:58 PM on February 16, 2022 [2 favorites]
I've always felt that the terminology is not terribly descriptive, I don't know why they are called "rings". But they are just sets of items where you can add, subtract, and multiply, but not necessarily divide. (The "addition" and "multiplication" are just operations which are required to obey a few obvious axioms).
The integers are the prototypical example. Another good example is sets of matrices, for example all 2x2 matrices form a ring where the "multiplication" is matrix multiplication. This is an interesting example because it's not commutative - for matrices A and B, AB is in general not equal to BA. The addition is required to be commutative, but not the multiplication.
Another good example is polynomial rings - consider for example all polynomials in a variable "x". You can add these and multiply them using standard algebra. So they are a ring. Similarly, you could consider all polynomials in several variables, say "x" and "y".
"Fields" are like rings but are one step better because you can divide (except for a special element, 0, which is not invertible). Real and complex number systems are fields.
These concepts are basically a big generalization and abstraction of things you are (probably) very familiar with, real and complex numbers, polynomials, and matrices.
posted by crazy_yeti at 4:06 PM on February 17, 2022 [3 favorites]
The integers are the prototypical example. Another good example is sets of matrices, for example all 2x2 matrices form a ring where the "multiplication" is matrix multiplication. This is an interesting example because it's not commutative - for matrices A and B, AB is in general not equal to BA. The addition is required to be commutative, but not the multiplication.
Another good example is polynomial rings - consider for example all polynomials in a variable "x". You can add these and multiply them using standard algebra. So they are a ring. Similarly, you could consider all polynomials in several variables, say "x" and "y".
"Fields" are like rings but are one step better because you can divide (except for a special element, 0, which is not invertible). Real and complex number systems are fields.
These concepts are basically a big generalization and abstraction of things you are (probably) very familiar with, real and complex numbers, polynomials, and matrices.
posted by crazy_yeti at 4:06 PM on February 17, 2022 [3 favorites]
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