Mathematics has the rock solid unit that doesn't change in series.
1 in 1 2 3 4 .....1325432 1325433 ....
But mathematics is a language, it's properties of the unit is singular, it abides by logical rules. There is nothing a about number has that says stop this is the limit.

In the real world,
units are multifaceted, multidimensional so interactions are physically limited by type and structure of the subunit. There are set ranges that certain sized subunits can be viable.

I proposed taking a census of all things and trying to examine them as discrete subunits. This census would range from subatomic particles, atoms, repeating crystal structures, micro-organisms, fauna and flora, geological structures, planet/ star sizes , galaxy sizes.

Does the universe go 1 2 3 .. or 1 2 7 9 12 13 14 15 21 ...*
Logically 5 exists but under construction rules of the universe not a magnitude subunit available.

The gaps might indicate the real world recursive physical patterns**.
The standard movie goes linearly from atom to web of galaxies but I not too sure that is the case.

Maybe pounds, shillings and pence was on the right track.

* Totally made up, to illustrate the idea.
** patterns that come as the small is repeatedly stacked to the large, Or as with life, the subunit pattern that is stuck between the very small and very large recursive patterns.

The Tukey Ladder of Powershttp://onlinestatbo...rmations/tukey.html For [pertinax] this is what I was crudely trying to describe - i.e. different degrees of curvature can suggest some underlying x^n relationship. Outside of this ladder of powers is the exponential relationship n^x which can look like an excessive version as n gets larger. [zen_tom, Jan 14 2021]

YouTube: 3Blue1Brownhttps://www.youtube...jab_esuFRV4b17AJtAw Regards maths content on YouTube, this channel has been really illuminating on a number of topics. Gives great perspective and context, whilst not shying away from (sometimes) quite technical detail. [zen_tom, Jan 14 2021]

E series of preferred numbershttps://en.wikipedi...f_preferred_numbers pick your level of precision, get a series of numbers [lurch, Jan 14 2021]

There are lots of transformations you can do that will convert from observed data to a linear pattern. For things that grow
monotonically with some degree of acceleration, this might crudely go from n^x, through a "ladder" of x^n forms (where n
goes from some positive integer through fractions down to large negative integers) with logx in the middle.

Then there are oscillating series, divergent or otherwise, and really interesting plays on complex series where rather than
the number line being on a line, it gets expanded into the plane, from where you get really special spaces like the Riemann
zeta function which both spirals chaotically, but also precisely spells out the locations of all the prime numbers, themselves
which have the property that their nth member tends to lie close to series defined by n log(n).

Other series of natural interest include the binomial series (itself generated or reducible from Euler's Gamma Function,
which itself is the analog of the factorial function applied to non-integers) which governs discrete combinatoric affairs* and
by extension a great deal of classic probability. Then there's Fibonacci series, and triangular series, each of which appear in
nature in different places.

Much of mathematics is astoundingly good at describing things we observe in nature, often doing so decades or centuries
before anyone is able to make any actual observations or even conceive of them. Much of our understanding of nature
today was explored mathematically long before it was ever observed.

Regards the idea of looking at the structure of things at different levels of scaling - that's been very closely studied under the name of
"scale invariance" and it's degree or strength (i.e. the amount of scaling you have to do to repeat a structure) is usually described as
being of some non-integer dimension. So if a 2d curve is somehow self-similar (i.e. it exhibits sub-units that look like macro-units) then
it will have an effective dimension of 2.x lying somewhere between 2 and 3. Different objects, systems and patterns exhibit different
levels of this dimension, so the coastline of the UK is estimated to be something like 1.2 other objects might be more, or less squiggly.

* re combinatoric affairs - I'd suggest this is where the secret sauce of nested structure lies - one atom alone isn't very interesting, but when there
exists a network of atoms all influencing one another through their network of interactions, you get the opportunity to create an interesting unit.
When multiple networks of these molecular units interact through their networks, again a transcendent unit is created. This network effect where
small, simple units interact across some network of relationships seems to be a common theme in nature, and again, there's a fair amount of
interesting mathematics that covers this phenomena. Unfortunately, the numbers involved in calculating the possible connections (permutations) of
a collection of n objects explodes into incalculably large values beyond relatively small collections, even say 20, after rounding produces something
like 2.4 million billion permutations, and the more you add, the faster the numbers get bigger - few computers can deal with such calculations,
even given time and lots of air-conditioning.

// Much of mathematics is astoundingly good at describing things we observe in nature, //

... except that due to neurophysiological structures, humans have an innate bias to "see" patterns - faces are the classic case - in what are actually truly random configurations, leading to the risk of self-fulfilling prophecies.

[zen], what you've written makes so much more sense than [wjt]'s original post- particularly about self-similarity persisting across micro to macro scales, as the Mandelbrot set demonstrates - that it's a pity that the two texts can't be exchanged ...

Someone should spend some time with my favorite
math channel gal, Vi Hart. Her explanations of
interesting math things are quite good.

Nature is less random than you think. Leaf growth
patterns are a function of simple growth hormone
maximal locations, which by their usage and
repetition make for some fascinating mathematical
patterns. See link.

I would just like to go on record to say that if it were possible to mentally download your understanding of the language of mathematics without having to go through all of the work it took for you to attain it... I would be good with that.

I understand patterns in nature have been found that have mathematical precis series or formulae like shell spirals and population interactions.

We all came from the expansion, how all the tiny space-time subunits unfolded/congealed to what we have now must also have a skeleton mathematical framework. Definitely not a even constant continuum.

And I don't think it is about repetition all the way through as combination effects can twist making a scale 'invariance' disappear.
If the scalar invariance is more about lowest/stable mass/energy ledges as scale increases , then this has been thought about, and I concede. Looking out into the world, there seems a lot of empty space with a lot of embroidery about the edges in blobs.

This more a general overview of whether matter/organization clumps, as scale increases, hence trying to shine a light the ultimate skeletal theory.

[kdf] This poor written concept, did get a well written, informed text from [zen_tom].

Experimental measurements are the constraints of the mathematical language.

//if it were possible to mentally download your understanding of the language of mathematics without having to go through all of the work it took for you to attain it//

Sounds great! But I think those two clauses are more intimately linked than you suspect... a bit like trying to get from America to Europe without crossing the ocean...

“...trying to get from America to Europe without
crossing the ocean...” —pocmloc, Jan 14 2021

Easy-peasy. Tunnel through the Earth. With good
vacuum pumps and gravity assist, trip times will be
90 minutes or less between any tunnel
entry/exit points. But allow some extra time for
getting
to and from the tube station and going through
security.

I'm a Vi Hart fan as well, and if people are looking for
other good
maths content, I'd recommend 3Blue1Brown who is
uniquely
good at explaining horribly *horribly* tricky material in an
intuitive, geometric way that in multiple cases has helped
me understand (or feel as though I understand) things
that
hitherto been completely alien. I'm still rewatching his
linear algebra series for maybe the 3rd or fourth time,
and
still finding it full of useful insight.

"The most effective place for a fixed link is the Bering Strait"
—8th of 7, Jan 14 2021

That's fine if you want to go shallow and only the short
distance. But in a vacuum tunnel straight through the Earth,
free fall gets you from where you are now to the direct
antipodes in about 84 minutes. And I think certain curved
paths might make other routes possible (e.g., New York to
London) in an even shorter time without going all the way
through the Earth's core.

On most of your planet, the antipode to any given position on land is quite likely to be ocean - easy enough to calculate with a bit of basic spherical geometry. So, wherever you move on land, you're likely to be crossing ocean, albeit with the thickness of the planet in between.

Unless, of course, you subscribe to the view that it's "turtles all the way down" ...

[wjt], if I'm understanding you correctly, then something you might be interested in looking at is called the "E series of preferred numbers". It's how resistors get their weird but consistent values.

I read the date of posting as "March". I'm going to pretend the post let someone make a time machine, and they went back in time to edit the post and keep anyone else from discovering it.

A coastline is more of part of subunit . Tetonic plates I would think might be a subunit but inter planetary data might be needed on that one. Each of the subunits is going to be physically controlled by multitude of variables and physical factors but all of those dimensions emerge from the ultimate machine's underlying metal.