Math Is Fun Forum

Please explain what is going on here
The big circle is fixed?
The mid-size circle ROLLS on it without slipping?
OR
Is attached to it at a fixed point - while the big circle rotates

The small circle rolls on the mid-size circle? At what rate?
OR
The small circle is attached fixedly to the mid-size circle (which spins?)
OR
The small circle spins - at what rate?

The pen is attached to a fixed point
ON the small circle
OR
Within the small circle
Or rotates around the circumpherence of the small circle - at what rate?

Many thanks

Is there a good way to simulate a 3-d plotter on a computer screen?

This question seems to be about bending of a bar.
But such a bar will bend under its own weight if horizontal or buckle if vertical
There is no limit to the ways it could bend depending upon
Which DIRECTION the one inch is - sideways or lengthwise.
HOW the bar ends are held  - clamped angle or free to pivot
How easily the steel of the bar compresses under axial force.

The bending moment at any position x of the bar is the sum of the axial forces times the y deflection at that point.  In this we should add any forces due to the weight of the bar.

A useful formula is the deflection of the bar at point x is
Bending moment M at point x  is E I d2ydx2 , where E is Young's modulus and I is the moment of area of the beam's cross-section at position x.

The simplest idea so far is this:-
For N>2 points, each point has a neighbour that is nearer than of all the others.
Lets make this distance as great as we can without going outside the sphere.
VERY SMALL increments are best!

We now REPEAT this for every single point in turn.
Now each point ends up max-spaced from its nearest neighbour

This is ONE variety of evenly-spaced.
But is it the only one?

To SEE this we need a way to visualise the pattern of points (on a flat screen)
For example for N=5 there is one pattern (of several?)
When we add a point we now have N=6 points.
If we could SEE how the points moved when we added that point it would help us begin to understand "what is happening" and why!

PLease suggest some ideas of how to visualise points scattered on or inside a sphere.
Many thanks

Thanks for your comments

So you are saying that "Truth" cannot be found by calculations.
You are saying it depends what sort of answer you want!

The sort of answer I'd find helpful is this:-
For the value of k that gives a stable 3-cycle from the iteration nextx=kx(1-x) do YOU say this is "sensitive to intital conditions" or not?
For the STRUCTURE is the same (a stable 3-cycle and an attractor) whatever tiny changes you make or suffer, for 0<x<1. Start in this range and you  stay in this pattern for ever - no escape!  If you start outside the 3-cycle range (but with 0<x<1) you soon get attracted into it

Not real planets!

The "mathematically pure" set of point-but-massive bodies moving relatively by Newton's Inverse Square Gravity Rule-of-Thumb without collisions.

There are a few position-velocity states into which, if placed, a third body will seem to be stable rotationg "motionlessly" with respect to the other 2.
A very-slightly-disturbed such body will move in an orbit about this so-called Trojan Point.
Are the calculated "strange attractors" imaginary artefacts due to computer error
How would we know?

The general question "How do I tell when my calculations are rubbish (even when totally reproducible) is most interesting and the opposite of chaos!
With chaos as nextx=4x(1-x) there is chaos spread throughought 0<x<1.
But for less than 4 there are interesting BANDS of permitted x and some periodic results.
The results are NOT evenly-distributed: their structure SEEMS stable against very small errors, but of course if we look at detail sufficient to "show errors" we will surely find them.

It is this ability to "accept and forgive" tiny errors while preserving the structural integrity for me to SEE - that is what fascinates me.

Guess the volts at the point where the series part joins the parallel part
Then find the currents in the 3 branches
When they add to zero you've got the volts right.
Successive guesswork

So for N=2,3,4,6,8.12,20,30 there is ONE best pattern of dots for each N
But for all other  N there are as many "best srrangements" as the number of ways we care to define "best even placement" - how many neighbours are at WHICH distance.

I was trying "Quick Reply" and after it had already acknowledged I was signed on
Quick Reply is denioed with idiot message endinmg  "Say when"

I was trying to send you a reference illustrating the vast difference between random and uniform
Random means irregular!

The ref is now lost and I have spent an hour trying to recover it - sorry abiut that.

To DENY crystalline's claim to be more uniform we need a way to MEASURE uniformity of spacing

Good idea!

By the way I like your "In mathematics, you don't understand things. You just get used to them." but for me it is "Just try them out!"

OK, so for a certain sphere, 1256200 points, the surface points nearest to one point vary depending on what we mean by nearest
Of ALL points only 2 (one pair) are nearest of all.
But for every point there are:-
A nearest point
A next nearest point
and so on

For example for point A its nearest dot is distance 3.12 away
The next nearest are 3.2,3.33,3.7 and 4

Is that "good"
What would be better?

For point B the nearest neighbours were
3.13, 3.19,3.3,3.73, 3.9

Is B better that A?
Might it pay to skip point B (or A) and replace it with a "better" point
How do we judge?

HOW do we begin to puzzle this out?

Great Stuff!

Can you add a way to halt the drawing so we can see how the wiggles are formed (the "stars" for example.

Did you find the wheel ratios by trial and error?