Creating a Fibonacci Spiral

[gwpriester]

My next series of stereograms on our eyeTricks 3D Facebook page will feature Fibonacci Spirals.

My ability to comprehend mathematical explanations is non-existent. So I decided to create a short video to illustrate how such a spiral is constructed.

The first square containing a quarter of a circle is duplicated and offset to the right.

From here on, the next square is the sum of the previous two squares. So 25px x 25px becomes 50px which becomes 75px, etc.

I discovered when making my spiral that for the best snap-to fit use the Square Caps setting for the quarter circle. It fits tightest into the square and does not extend outside the outline. My screen capture shows the new configuration of these settings inside the Selected Line Attributes drop down menu

The attached is called Fibonacci Yin & yang.

[Acorn]

Thank you Gary.

A Fibonacci sequence begins 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, ...

Like you I created a set of squares based on these dimensions, simply by matching a corner and dragging until I had a snap.
I created a Circle and from that a quarter circle.
I just cloned, scaled and rotated to fit into each box.

As an aside I tried to fit a Xara Spiral but while I could almost, the smaller turns were off.
I did what I could then Ungrouped the Spiral and manually matched the line to curves of the circles.

Instead, I took the Circles and and merged them into a Shape and converted a 1px Line width to a Shape.
Very carefully I then deleted or broke at points to recover an Fibonacci Spiral.
I cloned and scaled down and matched it with the inner end of the spiral and merged these two lines into one.

My workings: Fibonacci Spiral.xar

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Maths ahead

Take any two adjacent Fibonacci Numbers and divide the larger by the smaller.
55/34 = 1.618 approx.
For larger numbers this ration converges to the Golden Ratio, 1.618033988749894848204586834365638117720309179805 76286213544862270526046281890244970720720418939113 74847540880753868917521266338...

Its actual definition is phi = ( root(5) = 1 ) / 2.
Even phi = 2sin(54deg).
For Devil worshippers, phi = -2sin(666deg).
It continued fraction is (1; 1, 1, 1, 1, 1, 1, ...).

If you construction a Golden Ratio Spiral, it removed the quarter circle approximations.

Acorn

[gwpriester]

What I found interesting, was every time I added an additional square/quarter circle, I expected the number of twists to the spiral to increase.

They don't. It just makes the same simple spiral larger and larger.

[Acorn]

What I found interesting, was every time I added an additional square/quarter circle, I expected the number of twists to the spiral to increase.
They don't. It just makes the same simple spiral larger and larger.

No so Gary.

Think about it, if you add 4 quarter circles together you get a circle or one turn.
Even when the circles are increasing along with the Fibonacci Sequence. Your drawing area and monitor soon grow out of space.

The image here has boxes 0, 1, 1, 2, 3 (Turn), 5, 8, 13, 21 (Turn), 34, 55, 89, 144 (Turn), 233, 377, 610, 987 (Turn), 1597, 2584, 4181, ...



Carry on to the last two above and you are in 4k territory, starting with a single pixel and have managed all of 4¾ turns.

Acorn

[gwpriester]

I did one of golden section rectangle, spirals those several years ago.

When you explain things mathematically to me my eyes glaze over. It's what Mary says is like talking to the cat.

Before computers, in my advertising days, when I used to do my marker layouts to present our ideas to the client, I determined the center of something by making a tick mark at the start and one at the end on a piece of paper and then folding the page. The crease was the center.

[Acorn]

Gary, can you hand over to the cat please?

Your folding trick was only ever the middle of the straight line for start to end.
You ought to have used a map odometer and halved the total rolled distance.

Archimedes worked out the centre of gravity of things using weights and a balance.

Acorn

[gwpriester]

[Acorn]

Gary, the cat's been hitting the keyboard for too long; check his eyes!

So much for AI generated art. Creepy.

Acorn

[handrawn]

oh I like the cat... just my kind, as real ones make me sneeze, especially long haired ones....

[Acorn]

I mentioned the Golden Ratio as the convergent value as the Fibonacci Sequence gets larger.

The construction used by @Gary with squares can done with Golden Ratio Rectangles instead.
The quarter circle approach is that bit more accurate for all dimensions as the Golden Ratio Rectangle (GRR) is self-similar (a.k.a., a fractal).

My approach is to fit the largest GRR on the page, clone the GRR, rotate 90 degrees and shrink to fit along the narrow side.
Repeat until Xara chokes.

In my design file, I stopped at 0.9x0.6px as it was just about visible in a browser. I could have gone down to 10x6millipoints (one more full winding).

My biggest stumbling point was how to construct a GRR in the XDA.
TG came to the rescue with https://www.talkgraphics.com/showthr...529#post602529 with thanks to @Marco D.
I refined the approach so that everything just snapped to control points at every stage. This is given on the first page.

The second page is the Golden Ratio Spiral construction.

Golden Rectangle and Spiral.xar

@handrawn - despite all the GRRs, no cats (or even the elephant) were harmed in this exercise.

Acorn