Wikipedia:Reference desk/Archives/Mathematics/2024 December 24

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December 24

How did the Romans do engineering calculations?

The Romans did some impressive engineering. Engineers today use a lot of mathematical calculations when designing stuff. Calculations using Roman numerals strike me as being close to impossible. What did the Romans do? HiLo48 (talk) 05:50, 24 December 2024 (UTC)

The kind of engineering calculations that might have been relevant would mostly have been about statics – specifically the equilibrium of forces acting on a construction, and the ability of the design to withstand these forces, given its dimensions and the mechanical properties of the materials used in the construction, such as density, modulus of elasticity, shear modulus, Young modulus, fracture strength and ultimate tensile strength. In Roman times, only the simplest aspects of all this were understood mathematically, namely the statics of a construction in which all forces work in the same plane, without torque, and the components are perfectly rigid. The notion of assigning a numerical magnitude to these moduli and strengths did not exist, which anyway did not correspond to precisely defined, well-understood concepts. Therefore, engineering was not a science but an art, mainly based on experience in combination with testing on physical models. Any calculations would mostly have been for the amounts and dimensions of construction materials (and the cost thereof), requiring a relatively small number of additions and multiplications.  --Lambiam 19:03, 24 December 2024 (UTC)

Calculating with Roman Numerals might seem impossible, but in some ways it's simpler than our positional system; there are only so many symbols commonly used, and only so many ways to add and multiply them. Once you know all those ways you efficiently can do calculations with them, up to the limits imposed by the system.

And for a lot of things they relied on experience. Romans knew how to build circular arches, but rather than do calculations to build larger arches, or ones with more efficient shapes they used many small circular ones which they knew worked, stacked side by side and sometimes on top of each other. See e.g. any Roman aqueduct or the Colosseum. For materials they probably produce them on site or close by as they're needed.--2A04:4A43:909F:F9FF:FC7B:F1E8:19D6:124C (talk) 20:12, 24 December 2024 (UTC)

To add to the above, it wasn't really the Romans that were great innovators in science, engineering, etc.. I think more innovation and discovery took place in Ancient Greece and Ancient Egypt. Certainly Greece as we have a record of that. Egypt it's more that they were building on such a monumental scale, as scale no-one came close to repeating until very recently.

Romans were military geniuses. They conquered Greece, and Egypt, and Carthage, and Gaul, and Britannia, and everywhere in between. They then built forts, towns, cities and infrastructure throughout their empire. They built so much so widely that a lot of it still stands. But individually a lot of it isn't technically impressive; instead it's using a few simple patterns over and over again.--2A04:4A43:909F:F9FF:FC7B:F1E8:19D6:124C (talk) 12:09, 25 December 2024 (UTC)

See Roman abacus. catslash (talk) 22:03, 25 December 2024 (UTC)

It has to be said that in Roman times calculations for architecture were mostly graphical, geometrical, mechanical, rather than numeric. In fact, from the perspective of an ancient architect, it would make little sense translating geometrical figures into numbers, making numeric calculations, then translating them into geometrical figures again. Numerals become widely used tools only later, e.g. with the invention of Analytic Geometry (by Descartes), and with logarithms (Napier); and all these great mathematical innovations happened to be so useful also thanks to the previous invention of the printing press by Gutenberg --it's easier to transmit information by numbers than by geometric constructions. One may even argue that the invention of the printing press itself was the main reason to seek for an adequate efficient notation for real numbers (achieved by Stevinus). pma 21:22, 1 January 2025 (UTC)

This is a great answer! Tito Omburo (talk) 21:38, 1 January 2025 (UTC)

Architectural calculations, mentioned by Vitruvius, are not what I think of as engineering calculations. The former kind is about form. The latter kind should provide answers questions about structural behaviour, like, "Will these walls be able to withstand the outward force of the dome?" Can such questions be addressed with non-numerical calculations?  --Lambiam 23:40, 1 January 2025 (UTC)

Are these sequences mod any natural number n periodic?

The period of Fibonacci number mod n is the Pisano period of n, but are these sequences mod any natural number n also periodic like Fibonacci number mod n?

1. Lucas number
2. Pell number
3. Tribonacci number
4. Tetranacci number
5. Newman–Shanks–Williams number
6. Padovan sequence
7. Perrin number
8. Narayana sequence
9. Motzkin number
10. Bell number
11. Fubini number
12. Euler zigzag number
13. Partition number OEIS: A000041
14. Distinct partition number OEIS: A000009

42.76.153.22 (talk) 06:06, 24 December 2024 (UTC)

For 1. through 4., see Pisano period#Generalizations. Although 5. through 8. are not explicitly listed, I'm pretty sure the same argument applies for their periodicity as well. GalacticShoe (talk) 07:05, 24 December 2024 (UTC)

For 13, I'm not sure, but I think the partition function is not periodic modulo any nontrivial number, to the point that the few congruences that are satisfied by the function are also very notable, e.g. Ramanujan's congruences. GalacticShoe (talk) 07:23, 24 December 2024 (UTC)

It's possible that a sequences is eventually periodic but not periodic from the start, for example powers of 2 are periodic for any odd n, but for n=2 the sequence is 1, 0, 0, ..., which is only periodic starting with the second entry. In other words a sequence can be become periodic without being pure periodic. A finiteness argument shows that 1-8 are at least eventually periodic, but I don't think it works for the rest. (Pollard's rho algorithm uses this finiteness argument as well.) It says in the article that Bell numbers are periodic mod n for any prime n, but the status for composite n is unclear, at least from the article. Btw, Catalan numbers not on the list?. --RDBury (talk) 08:08, 24 December 2024 (UTC)

If only the periodicity of Bell numbers modulo prime powers were known, then periodicity for all modulos would immediately result from the Chinese remainder theorem. GalacticShoe (talk) 01:29, 29 December 2024 (UTC)

PS. 1-8 are pure periodic. In general, if the recurrence can be written in the form F(k) = (some polynomial in F(k-1), F(k-2), ... F(k-d+1) ) ± F(k-d), then F is pure periodic. The reason is that you can solve for F(k-d) and carry out the recursion backwards starting from where the sequence becomes periodic. Since the previous entries are uniquely determined they must follow the same periodic pattern as the rest of the sequence. If the coefficient of F(k-d) is not ±1 then this argument fails and the sequence can be pre-periodic but not pure periodic, at least when n is not relatively prime to the coefficient. --RDBury (talk) 18:12, 24 December 2024 (UTC)

Ah, what was tripping me up was showing pure periodicity, recursing backwards completely slipped my mind. Thanks for the writeup! GalacticShoe (talk) 18:31, 24 December 2024 (UTC)

good questions. What about TREE(n) mod k, for arbitrary fixed k?Rich (talk) 23:03, 27 December 2024 (UTC)

Link: TREE function. There are a lot of sequences like this where exact values aren't known, Ramsey numbers are another example. It helps if there is a relatively simple recursion defining the sequence. --RDBury (talk) 11:50, 28 December 2024 (UTC)

For 9. Motzkin numbers are not periodic mod 2. Motzkin numbers mod 2 are OEIS:A039963, which is OEIS:A035263 with each term repeated (i.e. $0\rightarrow 00,1\rightarrow 11$.) OEIS:A035263 in turn is the sequence that results when one starts with the string $1$ and successively maps $0\rightarrow 11,1\rightarrow 10$ (e.g. $110\rightarrow 101011$.) It is clear that if OEIS:A035263 were periodic with period $n$, then the periodic string $X$ of length $n$ would need to map to string $XX$, but this is impossible as the last character of $X$ is always the opposite of the last character of the map applied to $X$. Thus OEIS:A035263 is nonperiodic, and neither is OEIS:A039963. GalacticShoe (talk) 01:08, 29 December 2024 (UTC)

This paper (linked from OEIS) goes into more detail on Motzkin numbers. I gather the sequence might be called quasi-periodic, but I can't find an article that matches this situation exactly. A003849 is in the same vein. --RDBury (talk) 16:59, 30 December 2024 (UTC)

For 11., the article seems to suggest that Fubini numbers are eventually periodic modulo any prime power. I'm pretty sure this means that they the numbers eventually periodic mod any number $n$, since the lcm of the eventual periods modulo all prime power divisors of $n$ should correspond to the eventual period modulo $n$ itself, with the remainders being obtainable through the Chinese remainder theorem. However, the wording also seems to suggest that periodicity modulo arbitrary $n$ is still conjectural, so I'm not sure. GalacticShoe (talk) 02:44, 29 December 2024 (UTC)

You have answered all questions except 12 and 14, and 9 and 13 are the only two sequences which are not periodic mod n (except trivial n=1), 12 (Euler zigzag numbers) is (sequence A000111 in the OEIS), which seems to be periodic mod n like 10 (Bell numbers) (sequence A000110 in the OEIS) and 11 (Fubini numbers) (sequence A000670 in the OEIS), but all of these three sequences need prove, besides, 14 (Distinct partition numbers) (sequence A000009 in the OEIS) seems to be like 13 (Partition numbers) (sequence A000041 in the OEIS), i.e. not periodic mod n. 1.165.199.71 (talk) 02:27, 31 December 2024 (UTC)