A new visualization tool for OEIS sequences

[Dr George Whale]

Hello all, I've just joined SeqFan, and thought I'd introduce myself.

Let me confess first of all that I'm not a mathematician - my background's in art/design and software development.

A while back, I began developing an app to visualize character / number sequences as line drawings. I was using production (string rewrite) systems to generate sequences, but the results were mostly uninteresting due, I think, to limited variety in the output. Then I discovered OEIS!

The new version of the app (developed with my capable assistant, Claude.ai) transforms OEIS sequences into polyarc curve drawings. Some are chaotic-looking, others surprisingly structured - begging the question: do such visualizations tell us, at a glance, anything useful or significant about the structure and character of the underlying sequences?

The (free) app is called NCurve, and you can try it out here:
https://openprocessing.org/@GeorgeWhaleResearch/2986029
(Hint: press the 'New sequence' button repeatedly to search for interesting forms.)

Feedback (positive and negative!) welcome.

George Whale.

[Bill McEachen]

After fiddling with it a bit, my favorite so far is one added by Dr Sloane:

[George Whale]

That's a nice one. I might add a gallery of favourites alongside the app, if I can find a way to do it.

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[George Whale]

Hi Bill

I now have a small gallery, including your find: https://openprocessing.org/@GeorgeWhaleResearch/2986585

Best

George

[bill.m...@gmail.com]

Dr Whale:

Only becuase you wrote again, I attach other images I had found interesting (eye of the beholder) and saved off the other day.

Your gallery is great.

I cannot imagine the visualizations wouldn't provide useful insight in many cases.

I give "names" to most of them, and as you can see I iterated no parameters.

A376  French curve

A464  pie crust

A828  propeller

A1051  tire

A1553  saw blade

A1571 - ??

A1603 - ??

A39188  record disc

A39685  zipper

A39970  Slinky

Cheers

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[George Whale]

Thanks, these will help expand the library. I like your idea of giving them concise names, but with nearly 400,000 sequences in the OEIS, and the average vocabulary only around 25,000 words, it could become challenging!

Best

George W.

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[jpallouche.math]

Hi

For similar ideas and pictures in more ancient papers (in the 1980's), 
please see

F. M. Dekking, M. Mendès France, Uniform distribution modulo one: a geometrical 
viewpoint, J. Reine Angew. Math. 329 (1981), 143–153.
[cf. pp. 149 and 151]

J.-M. Deshouillers, Geometric aspect of Weyl sums, in Elementary and 
Analytic Theory of Numbers (Warsaw, 1982), Banach Center Publ., 17, 
PWN, Warsaw, 1985, pp. 75–82.
[available at https://bibliotekanauki.pl/articles/721340.pdf
similar pictures in all pages but the first]

Actually I did not have the time to check whether these two references
are somewhere in the OEIS, may be they are not.

best wishes
jean-paul

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[George Whale]

Bonjour Jean-Paul

Thank you so much for these references (from the 1980s!). I've been thinking of writing a short paper, and these will certainly help contextualise my efforts.

Best regards

George W.

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[Lisa Vitolo]

This is very cool, I've been looking at different sequences for an hour now :)

I'm thinking these would work nicely as wallpapers (for either laptops or smartphones), but I have to play with font size and aspect ratios a bit. And maybe add a black background and introduce more colours...

- Lisa

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The Cockroach King of guile and greed
With a broken crown he's left to bleed
An empire falling to its knees
A bleeding ground for those who heed

[George Whale]

If you Export SVG then you can change line weight, colour etc. (using a vector app such as Illustrator or Inkscape, which is free), even combine several drawings together.

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[DONG HAOXUAN]

Hi everyone,

I’ve been looking more closely at the mathematics behind NCurve and have found a few results that seem potentially quite interesting and relevant to the short paper you mentioned. I’m still checking and organising them, so I’d rather send you a clean write-up once that’s ready. If you’d be open to it, I’d be very interested in contributing to the mathematical side of the paper.

Kind regards,

Jason Dong

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[George Whale]

Jason, thank you for your kind offer, much appreciated! Would you be open to co-authoring?

I have in mind the Journal of Mathematics and the Arts: https://www.tandfonline.com/journals/tmaa20

What do you think?

Best

George W.

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[DONG HAOXUAN]

Hi George,

Absolutely — I’d be very happy to co-author this with you.

The results have actually turned out to be much richer than I expected when I first wrote. What began as a closer look at the mathematics behind the construction has developed into a fairly substantial theoretical structure, and I’m still finding new connections and some very stronger results as I push it further.

At this stage I’d rather not send you fragments, because the picture is still developing quite quickly. I’m continuing the analysis now and expect to have a strong, coherent set of results by this evening. I’m not yet sure how deep the final theory will go, but I should be able to give you a concrete account of what we have by tomorrow at the latest.

The Journal of Mathematics and the Arts certainly sounds like a natural home for the project as originally conceived. Given how much mathematical material is now emerging, though, I think it may be worth keeping the final choice of journal open until you’ve seen the results — the paper may end up being rather more mathematically substantial than either of us initially expected.

I’m very excited about where this is going.

Best,
Jason

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[George Whale]

In addition to a co-authored paper in JMA (suitable for non-specialist readers), of course I'd be happy if you wanted to write your own in-depth paper for a specifically mathematical journal.

Best

George W.

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[J.S. Seneschal]

Here are few results I got by modifying one of my favourite sequences (A000217) with terms from the sequence and the A-number itself.

Very fun tool - Spirograph for the 21st century!

[DONG HAOXUAN]

Hi George and Jean-Paul,

Thank you both very much — I really appreciate the encouragement and support.

I have to say, the mathematics has developed far beyond anything I expected when I first started looking at NCurve. The results emerging now are honestly quite astonishing to me. What initially looked like a relatively simple geometric construction has opened into a much deeper structure involving several different areas of mathematics, and the pieces are beginning to fit together in a surprisingly coherent way.

The investigation is still moving very quickly, so I want to let it run a little further before presenting anything prematurely. At this point, though, the amount and strength of the material are already well beyond what I had imagined at the beginning.

As soon as I have the results properly consolidated, I’ll let you both know immediately and send a clear account of what has emerged.

Thank you again — I’m extremely excited about where this is heading.

Best wishes,

Jason

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[George Whale]

Nice! I'll add some of these to the gallery.

You may be interested to know that (following a suggestion by Bill McEachen) I'm adding a text import feature to the app, so you'll be able to input the b-file of any OEIS sequence, and will no longer be limited to those provided in the pre-built blocks.

Best

George W.

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[Neil Sloane]

George, These are all great pictures!

Mabybe I missed something at the start of this thread, but could you say how you define the curve you get?  If the sequence is a(0), a(1), ..., what exactly is the equation to the curve?

The other thing I could not figure out:  say someone wants to study an OEIS entry that you haven't yet got in your gallery - when you go to the NCurve web page, how do ask it to draw say the curve for A123456?

[George Whale]

Hi Neil

Every curve is a chain of equal-length arcs connected end-to-end tangentially, where the (+ve or -ve) turning angle of each arc is derived from the corresponding sequence term, a(n), by:

degrees = a(n) mod b + c

b, c and number of terms drawn can be set interactively in the nCurve app to produce different curves from the same sequence.

(Press the 'i' button on the app for further info.)

Under each gallery image you'll see the sequence A-number and values of b and c used to create it, as well as no. of terms drawn.

Later today I'm adding an 'Import TXT' button to the app to enable import of any OEIS b-file, or any plain text integer list, or indeed any poem or prose passage (words converted to integers by mapping chars to ints and summating).

George

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[DONG HAOXUAN]

Hi everyone,

I thought I should send you an update, because the mathematics behind NCurve has developed much further than I expected — honestly, rather dramatically so.

When I first started looking at it, I was only trying to understand the drawing rule itself. Each sequence term, after reduction modulo (b) and addition of the offset, becomes the turning angle of an equal-arclength circular arc. So I rewrote the program as an exact piecewise-constant-curvature curve and began by studying fairly natural questions: endpoint formulas, periodic behaviour, and exact closure.

That already produced some interesting structure. Periodic residue words can be described by an (SE(2)) monodromy, and because the implemented angles are integer degrees, exact closure can be treated using cyclotomic algebra rather than numerical plots. We then found that, once the tangent closes, the displacement depends only on the places where the curvature actually changes. Long constant-curvature runs essentially disappear from the closure formula.

The real surprise came when I looked more carefully at the offset parameter.

Changing the offset is exactly a frequency modulation of the unit tangent signal. At a natural lattice of offsets, the final NCurve endpoints are literally the Fourier coefficients of that tangent function.

That was the point where the project suddenly became much larger than I had expected.

The high-frequency spectrum is controlled by the curvature-changing junctions, and the same junction quantities also appear geometrically in the discrete evolute formed by the centres of the circular arcs. This gives an exact Fourier/evolute correspondence rather than just an analogy.

The next unexpected step was to encode each curvature change as a phase-weighted incidence vector. Once that is done, the Fourier covariance of these incidence vectors turns out to be exactly the Laplacian of the multigraph whose vertices are the active curvature values and whose edges record curvature transitions.

So, rather unexpectedly, the path became

NCurve geometry
→ Fourier analysis
→ curvature junctions
→ incidence data
→ a transition graph.

From there the graph-theoretic structure opened up very quickly. A Cauchy/resolvent version of the incidence spectrum recovers the whole transition Laplacian, which means that spanning trees and forests, effective resistance, vertex separators, and the magnetic/holonomy version of the graph all become accessible directly from the NCurve spectral data.

The most recent result is probably the one that surprised me most.

Suppose the active curvature values are
[
\theta_1<\cdots<\theta_d
]
and define
[
P(x)=\prod_{r=1}^{d}(x-\theta_r).
]

If the (d-1) critical points of (P), i.e. the roots of (P'), are used as resolvent probes, their normalized resolvent vectors form an exact orthonormal basis of the Dirichlet space of the transition graph.

Consequently the probe Gram matrix and the transition Laplacian are related by an exact isometry. The tomography condition number is exactly

[
1,
]

which is the theoretical optimum — not an asymptotic or numerical optimum.

That result has more or less closed the graph-tomography part of the problem: the number of real probes is sharp, the optimal probe locations are explicit, the reconstruction is exact, and the integer transition multigraph even has a dimension-independent decoding margin after normalization.

So what began as “what mathematics is hiding behind this sequence-drawing program?” has somehow turned into a fairly coherent chain

piecewise-curvature geometry
→ exact closure
→ Fourier-offset theory
→ evolutes and junctions
→ incidence spectra
→ graph Laplacians
→ resolvent/forest theory
→ optimal spectral graph tomography.

I am stopping at this point to organise and independently check the results properly, because there is now enough material that continuing to accumulate theorems without consolidating them would probably be a bad idea.

There are still some genuinely open parts. In particular, the upstream problem of recovering the incidence/Zak data stably from raw NCurve endpoint samples is not finished, and the behaviour of the current finite offset window is still producing some interesting extremal examples.

But the part from the resolvent/incidence data to the transition graph now looks surprisingly complete.

I thought you might enjoy seeing how far this has gone, especially since it started from such a simple visual construction. I certainly did not expect to end up anywhere near spectral graph tomography when I first opened the code!

I’ll be very happy to send you a more organised note once I have finished consolidating the proofs and references.

Best,
Jason

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[J.S. Seneschal]

I'll be looking forward to that TXT feature as there were some other sequences I wanted to play with but could not access. Speaking of which I did a more thorough dive into A000217 and found many more interesting shapes.

This first set comes from calibrating Show and (b) to the triangular number closest to their limits (1953, 351), then for (c) alternating between - and + for the first 15 terms of A000217:

This next set follows the same process, except using 217 for (b):

Finally, some random triangular and circular shapes derived from various other triangular terms:

[George Whale]

This looks fascinating, Jason - (with my limited maths) I will work through it very slowly to understand your insights.

My interest in visualization is that it can reveal patterns that other approaches miss - but of course it takes a domain expert to recognize their significance.

Best

George

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[George Whale]

Thanks, I added the triangular set to the gallery https://openprocessing.org/@GeorgeWhaleResearch/2986585, but didn't add the others as the numbers were too small / blurry. (It's useful to have readable numbers in case others want to reproduce the shapes.) The 3 x 3 layout (around 2000-2400px wide) works best, I think.

P.S. I've added the b-file/text import feature to the app, so now you can import any OEIS sequence via its b-file. https://openprocessing.org/@GeorgeWhaleResearch/2986029

George W.

[J.S. Seneschal]

TXT feature works great - using it right now!

PS. I had to upload smaller images because the file sizes were too big to post all at once. 

I have the full size versions, too, if you ever want me to send them to you.

[George Whale]

If you get a moment, it would be nice to have 3 x 3 of the circular ones in a single 2400 x 2400 image.

Best

George

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[Joshua Weinstein]

Hello everyone,

I found another sequence A049651 that seems to be able to generate a lot of cool structures. Generally speaking there seems to be some sequences such as this one 
and the one used by J.S. Seneschal that "just work" and create a lot of nice symmetrical patterns for different values of b and c. Overall I think Ncurve is super cool and I am excited to see all the interesting patterns that emerge.

Here are some of the cool patterns I found from A049651: https://drive.google.com/drive/folders/1LNj2S54wmOkBaW3Q4n0S_gOEv6mO8DBX

I will probably add more as I keep playing with it.

Best regards,

Joshua Weinstein

[J.S. Seneschal]

Here's a selection of what I thought were the 9 most aesthetically interesting. Let me know if you wanted a different set of 9...

... and here's a further set squaring the circle - or, rather, "circling the squares" A000290:

  

[J.S. Seneschal]

And, while I'm at it, here's some 4-fold squares (A000290), some 5-fold pentagonals (A000326), and some 6-fold hexagonals (A000384):

After playing with a bunch of sequences now I've found that once you get a hold of the right mod number, it's almost impossible not to make a series of interesting shapes! 

[George Whale]

J. S., Joshia, thanks for sharing your images - I hope you don't mind, I've added a number of my favourites to the gallery https://openprocessing.org/@GeorgeWhaleResearch/2986585

Best

George.

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[Joshua Weinstein]

George, I don't mind if you add any of the media I share. After using the website for a while here are some features I think would be nice to add down the line: 

1. It would be cool if we could choose where we want to start in the sequence. Maybe I want to start drawing from the 2nd term of a sequence or at the 638th term in the case of https://oeis.org/A133058, as the sequence starts off kind of randomly and then becomes quasi-periodic which can be seen in the 2nd example.

 

2. People have already mentioned choosing the color of the image which you already said can be changed by saving the SVG file, but I think there might be merit in being able to oscillate between colors. For example maybe I want every first curve to be blue every 2nd to be red and every 3rd to be green. Or maybe the first 20 blue then the next 20 green etc. This might help visualize patterns more and maybe we can get each corner of some shapes to be different colors. Also I'm not very familiar with SVGs so if there already a way to do this I apologize.

3. The ability to animate curves as terms are added or the b/c changes would be nice. It could be useful to see how the curve is drawn in addition to the final product and sometimes interesting patterns emerge when the b or c changes such as in the video here where you can see a shape similar to an Euler spiral emerge at around 23 seconds as the value of b changes. It would also be nice to be able to change the increment of terms added so maybe I can add 10 terms at a time instead of just 1. You can kind of already make animations of 1 term at a time if you hold your mouse on the up arrow and screen record such as this, but it would be nicer if we could just have 1 term = 1 frame and then change the FPS in a video editor or something.

I could try to modify your code and add these things on my own of course but I don't know HTML so I probably won't make much progress, if any. Let me know what you think overall though. Also I want to emphasize the app is already great as is and these extensions would just make it even better.

 Best regards,
Joshua

[J.S. Seneschal]

I don't mind at all either - in fact, I'm trying my best not to bombard you with dozens of gallery contributions! Just a few more for now...

It's interesting to observe the various genera of re-occurring shapes, such as the sinuous spirals in Joshua's set for A049625, or wavy/wobbly lines in his A019431 set usually produced by negative (c) numbers. Two others I've noticed are the "fancy filigree" type, as exemplified by the following set from A002378, mod 342; and the highly geometric forms exemplified by A373733, mod 55 below it:

[J.S. Seneschal]

...A373733 also demonstrates the vast differences that can be produced by changing mod (b) by just 1 number (in this case 150 vs 151):

One final contribution for now, a set of "figurate figures", some vaguely humanoid shapes found in the the polygonal figurate numbers from triangular (A000217) to nonagonal (A001106):

[George Whale]

Hi Joshua

Thank you, these are all great suggestions, and I will try them out with a view to possibly incorporating them.

Though I'm an experienced coder myself, since my nephew showed me how to use AI for development, I've barely written any code - instead I write prompts telling the AI what to do. This is both sad (having spent all those years learning!) and fantastically liberating.

Since NCurve and its code are free to use and modify for personal or educational purposes, anyone is free to play with it and adapt it if they want to.

I use Claude.ai but there are quite a few options out there now. It's really a lot of fun!

Best

George

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[George Whale]

Thanks J. S - 3 added.

[Joshua Weinstein]

Hi George, I took your advice and used Gemini to vibecode the features that I wanted. It honestly worked surprisingly well and you can try it out by copying the code from this google doc and pasting it into the code section of your website. I'll admit the colors can sometimes look harsh and I mostly prefer having only black but it can be cool to have color for some patterns. Also one more thing my new favorite sequences to generate patterns of is the Recamán's sequence where you can really see the chaos and order clashing. Attached below is one example but there are more examples in this drive.

Best regards,
Joshua

[George Whale]

Thanks Joshua, I will definitely have a play with your colour version.

I too prefer drawings that hover / alternate between order and chaos, see e.g. the A039516 set in gallery.

I googled 'Recamán's sequence' and its Wikipedia page https://en.wikipedia.org/wiki/Recam%C3%A1n's_sequence contains a more formal arc-based visualization - very nice.

George

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[Ed Pegg]

Now that I've seen a few of these....
are they related to the 1995 Clifford Pickover Curlicue fractals?   https://paulbourke.net/fractals/curlicue/  
https://mathworld.wolfram.com/CurlicueFractal.html  

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[jpallouche.math]

Hi

Curlicues! that is *the* word. Well done!
Searching this word in titles of papers on mathscinet
gives 7 results, including the nice paper by Moore and
van der Poorten available at 
https://maths.anu.edu.au/files/CMAProcVol22-MoorePoorten.pdf

best
jean-paul

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[George Whale]

Thanks Ed and Jean-Paul, a very interesting precedent (which, needless to say, I was blissfully unaware of!) and thanks for the links to papers.

The method I employ in NCurve is very similar to Pickover's in its chaining of equal-length elements (my arcs, his line segments) differing in angle (my turning angle, his orientation).

The main difference, I guess, is that his angle sequences (as I understand) derive from one formula, producing particular characteristics (notably self-similarity), whereas NCurve's draw from the huge diversity of OEIS sequences (I claim that as my original contribution!), so its potential output is, possibly, much more varied.

But yes, well spotted.

George.

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[George Whale]

Joshua, some of those latest examples are bizarre - but very striking! I'll put a few in the gallery.

Best

George

[Ed Pegg]

So far as I know, the curlicue fractal behavior isn't well-understood... and the new method adds complexity.... but it may be that the addition leads to better explanations.

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[George Whale]

Hopefully, seeing more of these sometimes surprising patterns might prompt people to look at them more deeply.

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[J.S. Seneschal]

Those are some wild shapes! And the wildness definitely seems to benefit from the large number of terms. I found some more restrained forms of "controlled chaos" the other day in A000867 - "Numbers beginning with letter 'f' in English." I suspect, however, the chaos is only apparent, and more structured shapes would arise from many more terms:

On the other side of the ordered chaos coin, A000819, mod 108, produces these very simple shapes in what looks like an almost hand-drawn fashion: 

Whereas A049094, mod 21, produces this twisted-up Slinkys:

[J.S. Seneschal]

...and two more for the road: odd patterns from A000494, "Nearest integer to sin(n)", and strange circles from various sequences:

[William Parker]

Long time lurker…sorry to ping everyone with this - but this tool is very fun. 

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[George Whale]

Thanks for these J. S. - posted: https://openprocessing.org/@GeorgeWhaleResearch/2986585

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[Neil Sloane]

Concerning George Whale's visualization tool, can someone explain what the pictures tell you about a sequence? What exactly are they good for?

Best regards

Neil 

Neil J. A. Sloane, Chairman, OEIS Foundation.

Also Visiting Scientist, Math. Dept., Rutgers University, 

Email: njas...@gmail.com

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[Simon Plouffe]

Hello, 

 I am certain of the importance of these graphics but
would it be too much ask ; not to re-re-re send
 the 7.2 megs file each time you reply to the list, 
there has been 100 replies to this message, it would
be a good idea to just reply to the last message and
 not send all the messages so far on top of the list + 
the 7.2 megs files each time, 
 just my 2 cents advice, 
 best regards, 
 Simon Plouffe

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[Jerry]

Yes, please!

[Joshua Weinstein]

Hi all,

In terms of what the images mean, I don't know if you can extract a ton of information from looking at one image alone, but I am pretty sure if you encounter a symmetric pattern of any kind or a chain of the same connected shape over and over it means the sequence you are looking at has a periodic modulo residue for your b value. If you let S be the sum of one period of your residue, the number of symmetry folds is 360/gcd(S, 360) and the number of terms to close a loop is P*360/gcd(S, 360) where P is the period length of the modulo residue. However, if S == 0 mod 360 and there is non-zero positional displacement across the period, the loop will not close and instead form an infinite chain of the same shape.

Best regards,

Joshua Weinstein

[Allan Wechsler]

Periodicity in all moduli is one of the hallmarks of sequences arising from linear recurrences.

-- Allan

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[jpallouche.math]

Dear all

Another keyword worth searching is "Turtle", see, e.g.,

H. Zantema. Turtle graphics of morphic sequences. Fractals, 24(1), 2016.

preversion available at http://www.win.tue.nl/∼hzantema/turtle.pdf.

H. Zantema. Playing with Infinity: Turtles, Patterns and Pictures. CRC

Press, Taylor & Francis group, 2024. 226 pages.

best wishes
jean-paul

[jpallouche.math]

The link below does not work. Replace it with
https://hzantema.win.tue.nl/turtle.pdf

best
jp

[Dr George Whale]

Dear All

First, thanks very much to those who shared links to research papers - v. interesting.

Second, I've added some enhancements to NCurve, including:
- double-handled slider to set exact sub-sequence to be drawn
- curve colouring option
- animation button
(Thanks to Joshua Weinstein for suggesting these.)
Also:
- medium / high-resolution PNG export options
https://openprocessing.org/@GeorgeWhaleResearch/2986029

Third, a couple more apps:

1. NCurvePlus enables you to combine (in simple expressions) two sequences and draw the result. I don't know if this adds anything useful, but it's quite nice to play with:
https://openprocessing.org/@GeorgeWhaleResearch/2993562
Here are some images I made earlier with it:
https://openprocessing.org/@GeorgeWhaleResearch/2993560

2. NCheck enables you to design check textile patterns from sequences (it's a bit of fun!):
https://openprocessing.org/@GeorgeWhaleResearch/2989873

Best

George W.

[J.S. Seneschal]

Fantastic new features!

Had to contribute a few sets after some initial explorations....

Some colourful floral arrangements from A028724:

Increasing disorder from A002378: