I composed this shape (in 2019) by overlapping decagons, a technique that was employed by Robert Reid in the book ‘The Gentle Art of Filling Space’. However, my motif generating procedure differs, in that the decagon overlaps occur at midpoints between decagon vertices as opposed to Robert’s coincident vertices (thank you to Michael Dowle for pointing that out).

I later stumbled across a similar shape, discovered by Tim Lexen, that uses only three arcs, called a tricurve. Both the penguin and tricurve can tile in the same way but according to Tim, the penguin has more tiling possibilities.

http://paulbourke.net/geometry/tricurves/

https://aperiodical.com/?s=tricurves

I like the tricurve for its smoothness but I find the penguin better to work with, as it has points of reference, such as edge midpoints and vertices.

Penguins can be put together in a number of ways.

A typical radial tiling.

Two-fold rotation.

A random selection of central cores that can expand forever.

The beginnings of a single tailed zigzag spiral.

Modified central cores.

Tilings with holes. Small and large decagons and octagons.

An attractive decagram and five pointed star.

Just found this five tailed spiral. I’m quite sure there are others.

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To David Smith: In your Penguin Nonagons, I was very impressed with the creativity, the wide range of designs you were able to create with the Penguin shape(s). From my standpoint they are faceted versions of tricurves. But with the Penguin shape you can do things impossible with tricurves, such as putting together two sections than in tricurves would both be concave or or both convex and thus not fit. In addition to images on Paul Burke’s sites, there are several articles on tricurves at https://aperiodical.com/?s=tricurves, as well as lots of images and other articles if you just Google “tricurves”. One phenomenon unique to tricurves is the existence of phantoms, as shown in articles at https://aperiodical.com/2018/10/phantom-tiling/ and https://plus.maths.org/content/ghosts-tiles. If you are interested in discussing these or other ideas please contact me at novustcl@charter.net. Best Wishes, Tim Lexen

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Hello Tim, thank you for your comments.

I had assumed (wrongly it seems) that the penguin and tricurve would work in exactly the same way. I will look into this and update the blog accordingly. So, did you discover the tricurves, as I’d also like to give you a mention? Update: upon further reading, I found out that you did discover tricurves!

There were a couple of interesting articles that I had read previously and left a comment in 2019 (which you replied to); https://aperiodical.com/2019/02/making-tricurves/ and https://aperiodical.com/2018/09/combining-tricurves/

Phantom tiling is new to me; not quite figured it out yet.

Kind regards – Dave S.

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Hi David, I started a reply to you, then either lost it or accidently sent it –so please excuse possible duplication. Thanks for your comments in 2019 — sorry did not recognize your name. Yes I did discover, develop and name tricurves many years ago, but did not start sharing and getting feedback online until 4 years ago (2017). Paul Burke was the first one to respond and make his own designs, and share thoughts on his website. Since then many others have encouraged, designed, contributed and shared. One recent one is a man in Japan I’m in touch with; you can see some of his designs if you scroll down at http://www.lcv.ne.jp/~hmika/memo/m20_10b.html.

Yes I also have assumed that a tricurve and its faceted version would tile exactly the same. But then I was looking at shapes where all the facets had the same lengths, which is different from your penguin design. By the way, with either penguins or tricurves, it seems the designs with 36- or 72-dgree corners (allowing 5- and 10-fold symmetry) are the most interesting. Hope you keep on experimenting!

With Best Wishes, Tim Lexen

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Hi Tim, yes I do that sort of thing, where emails get accidentally sent, lost or sit in the drafts folder unsent.

Thanks for the confirmation that you had discovered the tricurve. I did edit my post yesterday to include that information but it was seemingly too late.

I took a look at the link you gave me. I particularly like the way he combines mirror image tiles to create kite shaped arrangements.

Need to get on with some work in the garden now whilst the sun is shining!

Thank you for your interest.

Kind regards – Dave S.

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Hello David, I would like to send you some material by email, on newly-developed aspects of tricurves that you might find interesting. Could you please send me your email address? Please email to novustcl@charter.net. Thanks, Best Wishes! Tim Lexen

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