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The sicence behind 1 to 4 to 9 precision (2001 a Space Odyssey)


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Not really spoilers, but I'm just being safe if you haven't read the book or watched the movie:

Spoiler

In 2001 a Space Odyssey, the black monoliths have sides that extend into precise 1 4 9 ratios, and in the book, it said that's it's theoretical.

 

Edited by Spaceception
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This ratio has no real significance in engineering terms. I don't know why people would consider it the "best" apart from perhaps aesthetic reasons. In reality, a tensile structural member should generally be a circular section to minimise stress concentrations at corners, and a compressive or bending member should be something like an "I Beam", which maximises second moment of area and buckling load.

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9 minutes ago, peadar1987 said:

This ratio has no real significance in engineering terms. I don't know why people would consider it the "best" apart from perhaps aesthetic reasons. In reality, a tensile structural member should generally be a circular section to minimise stress concentrations at corners, and a compressive or bending member should be something like an "I Beam", which maximises second moment of area and buckling load.

Well, it is a book written in the 60s.

Edited by Spaceception
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There is no real significance in it being a sequence of squares of primes integers. What was significant was that a) so far as the characters could measure the proportions were perfect, ie the builders had extremely precise manufacturing process, and b) the numbers were relatively easy for readers to understand, unlike (for instance) pi, e and other irrational numbers.

Edited by softweir
Slight rewording. Plus correction - thanks Razark!
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2 minutes ago, softweir said:

There is no real significance in it being a sequence of squares of primes. What was significant was that a) so far as the characters could measure the proportions were perfect, ie the builders had extremely precise manufacturing process, and b) the numbers were relatively easy for readers to understand, unlike (for instance) pi, e and other irrational numbers.

Irrational in 10-based system ;) 10 is not always ten...

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An irrational number is irrational in any base. They can have a neat representation in irrational bases, but just because pi is 10 in base pi doesn't mean it's no longer irrational.

Edited by andrewas
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Quote

Irrational in 10-based system

The irrationality has nothing to do with the decimal (or any other) representation. It is to do with the fact that it is not equal to any rational number. You define rational numbers without even talking about 'base' anything.

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1 hour ago, andrewas said:

An irrational number is irrational in any base. They can have a neat representation in irrational bases, but just because pi is 1 in base pi doesn't mean it's no longer irrational.

thats a interesting idea! i read books with aliens that use base six, and the more familiar base 8, 2 and 10. i have never considered that the unit of 1 could mean one of anything. imagine an entire mathematical system built around pi=1. even better, have the orders of magnitude non-linear according to the inverse square rule...

thats some alien math. i wonder what kind of world view would be required for a race to develop something like that?

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Yep, and you can't count in base pi. It just doesn't work.

1 is pi^0

10 is pi^1

100 is pi^2

What is 3? What is 9? If your base isn't an integer, numbers that aren't 1, 10, 100, 1000 etc don't make any sense. If you divide the interval up into chunks, you're no longer working in base pi, you're just giving another name to an integer base system.

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1 hour ago, peadar1987 said:

Yep, and you can't count in base pi. It just doesn't work.

1 is pi^0

10 is pi^1

100 is pi^2

What is 3? What is 9? If your base isn't an integer, numbers that aren't 1, 10, 100, 1000 etc don't make any sense. If you divide the interval up into chunks, you're no longer working in base pi, you're just giving another name to an integer base system.

Have a look at these entries. Turns out non-integer bases have been contemplated and sometimes even used.

https://en.wikipedia.org/wiki/Non-integer_representation

https://en.wikipedia.org/wiki/Ostrowski_numeration

We need to draw a sharp line between the symbols themselves and what values they represent, which in turn gives them their properties.

 

EDIT: Here is one interesting read

http://www.americanscientist.org/issues/pub/third-base/2

Edited by LN400
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9 minutes ago, LN400 said:

Have a look at these entries. Turns out non-integer bases have been contemplated and sometimes even used.

https://en.wikipedia.org/wiki/Non-integer_representation

https://en.wikipedia.org/wiki/Ostrowski_numeration

We need to draw a sharp line between the symbols themselves and what values they represent, which in turn gives them their properties.

 

EDIT: Here is one interesting read

http://www.americanscientist.org/issues/pub/third-base/2

Yeah, I think it's pretty clear how you'd define such a system if you had to (multiplying by powers of the base and so on). But does it make sense when you use it? Let's count in base π.

1, 2, 3, 10, 11, 12, 13, 20, 21, 22, 23, ...

Looks a lot like base 4, except the numbers aren't evenly spaced; 10-3 ≠ 3-2. And while '3' and '1' are valid non-fractional symbols by themselves, '3+1' can't be represented without fractions. I think that's what @peadar1987 was objecting to.

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4 hours ago, softweir said:

There is no real significance in it being a sequence of squares of primes.

With the sequence 1 4 9, you can't really say it's the squares of primes.  It could simply be the squares of integers.  You'd need to know if the next element of the sequence is 16 or 25.

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4 minutes ago, HebaruSan said:

Yeah, I think it's pretty clear how you'd define such a system if you had to (multiplying by powers of the base and so on). But does it make sense when you use it? Let's count in base π.

1, 2, 3, 10, 11, 12, 13, 20, 21, 22, 23, ...

Looks a lot like base 4, except the numbers aren't evenly spaced; 10-3 ≠ 3-2. And while '3' and '1' are valid non-fractional symbols by themselves, '3+1' can't be represented without fractions. I think that's what @peadar1987 was objecting to.

The problem I see here is, you are not really counting in base pi here. It is a pure base 4 in symbols but not in value. The problem arises from what values should the symbols represent, and it is here you jump back and forth between base pi and base 4, thus getting weird results. Same was with peadar's post, pointing at a base pi system while keeping the base 10 values for the symbols.

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You would try with the golden ratio (φ = (sqrt(5) + 1) / 2 ~= 1.618... )

As:
φ-1 = φ - 1
φ0 = 1
φ1 = φ
φ2 = φ + 1,
φ3 = φ2 * φ = (φ + 1) * φ = φ2 + φ = 2φ + 1,
etc

Edited by kerbiloid
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5 hours ago, Spaceception said:

(...) in the book, it said that's the theoretical best ratio you can have in construction (...)

I’ve read the book many times, although a long time ago (before the year 2001 at least) but I don’t ever recall something about the notion that 1:4:9 is the best ratio for construction. Perhaps you misinterpreted a phrase?

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45 minutes ago, LN400 said:

The problem I see here is, you are not really counting in base pi here. It is a pure base 4 in symbols but not in value. The problem arises from what values should the symbols represent, and it is here you jump back and forth between base pi and base 4, thus getting weird results. Same was with peadar's post, pointing at a base pi system while keeping the base 10 values for the symbols.

Well, what else should I do? We don't switch to new symbols or change their meaning (in the ones column at least) for base 2 or 4 or 8, and for base 16 we just need to add a few new ones. What would be the correct symbols to use for base π?

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1 minute ago, HebaruSan said:

Well, what else should I do? We don't switch to new symbols or change their meaning (in the ones column at least) for base 2 or 4 or 8, and for base 16 we just need to add a few new ones. What would be the correct symbols to use for base π?

That is a real good question. With 1 we think of well, one this or one that, a single unit of something and with 3 we think of a single unit together with a single unit together with yet another single unit for a collection of well, 3 units. That works well with base [any integer] system but it doesn't necessarily work well if the base in the new system is an irrational number or worse, a transcendental number.

If the base is an irrational number then we would need a system that is consistent and non-self contradictory, a system that allows some kind of operation according to a set of rules. It is in that sense not strictly necessary to have the numbers evenly spaced. Think about logarithms and how spacing is non-even.

I do not know how one can do it but one thing to ponder, consider complex numbers. Even if we stick with the familiar digits, the real number 1 in the complex plane would be 1 + 0i. This isn't just 2 values added together, it IS the "symbol" for 1, in cartesian form that is. Writing it as 1 is more of a shortcut not writing down what we have none of (the imaginary part). Perhaps it would be needed to give the values in a say, base pi system a whole different breed of symbols than we are familiar with.

Then again, I'm no great matematician so what do I know :D

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Correction to my earlier post - pi is 10 in base pi, not 1. 

As for counting in base pi, no integer above pi has a neat representation. 1 is 1, 2 is 2, 3 is 3, 4 is 10.22.... and I can't work out any more digits on the phone while still inpretending to listen to this conversation.

It is not a practical base for everyday use, but it probably has applications somewhere.

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1 hour ago, Kerbart said:

I’ve read the book many times, although a long time ago (before the year 2001 at least) but I don’t ever recall something about the notion that 1:4:9 is the best ratio for construction. Perhaps you misinterpreted a phrase?

Quote from page 173 from the paperback, Chapter 31, Survival;

Quote

One curious, and perhaps quite unimportant, feature of the block had led to endless argument. The monolith was 11 feet high, and 1 1/4 by 5 feet in cross-section. When it's dimensions were checked with great care, they were found to be in the exact ratio 1 to 4 to 9--the squares of the first three integers. No one could suggest any plausible explanation for this, but it could hardly be a coincidence, for the proportions held to the limits of measurable accuracy. It was a chastening thought that the entire technology of Earth could not shape even an inert block, of any material, with such a fantastic degree of precision. In its way, this passive yet almost arrogant display of geometrical perfection was as impressive as any of TMA-1'S other attributes.

So I did misinterpret it a bit. But (I don't think) I didn't misinterpret 100%.

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5 minutes ago, Elrond Cupboard said:

1 isn't prime.

I've seen it argued both ways.  I thought about bringing it up in my own post, but ended up not doing so.

Of course, with the above post actually quoting it, the argument is pointless.  It does say integers, and not primes.

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Just now, razark said:

I've seen it argued both ways.  I thought about bringing it up in my own post, but ended up not doing so.

Of course, with the above post actually quoting it, the argument is pointless.  It does say integers, and not primes.

Hey, you're in the 1000 rep group :)

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