{"id":626,"date":"2009-02-07T16:48:57","date_gmt":"2009-02-07T15:48:57","guid":{"rendered":"http:\/\/www.walkingrandomly.com\/?p=626"},"modified":"2010-12-21T11:56:04","modified_gmt":"2010-12-21T10:56:04","slug":"quadraflakes-pentaflakes-hexaflakes-and-more","status":"publish","type":"post","link":"https:\/\/walkingrandomly.com\/?p=626","title":{"rendered":"Quadraflakes, Pentaflakes, Hexaflakes and more"},"content":{"rendered":"

Here in the UK we have had more snow than we have seen in over 20 years and as a country we are struggling with it to say the least.\u00a0 I have friends in places such as Finland who think that all this is rather funny…it takes nothing more than a bit of snow to bring the UK to its knees.<\/p>\n

Anyway…all this talk of snow reminds me of a Wolfram Demonstration<\/a> I authored around Christmas time called n-flakes.\u00a0 It started off while I was playing with the so called pentaflake<\/a> which was first described by someone called Albrecht D\u00fcrer (according to Wolfram’s Mathworld).\u00a0 To make a pentaflake you first start of with a pentagon<\/a> like this one.<\/p>\n

\"pentagon\"<\/p>\n

Your next step is to get five more identical pentagons and place each one around the edges of the first as follows<\/p>\n

\"pentaflake<\/p>\n

The final result is the first iteration of the pentaflake design.\u00a0 Take a closer look at it….notice how the outline of the pentaflake is essentially a pentagon with some gaps in it?<\/p>\n

\"pentaflake<\/p>\n

Lets see what happens if we take this ‘gappy’ pentagon and arrange 5 identical gappy pentagons around it – just like we did in the first iteration.<\/p>\n

\"pentaflake<\/p>\n

The end result is a more interesting looking gappy pentagon. If we keep going in this manner then you eventually end up with something like this<\/p>\n

\"pentaflake\"<\/p>\n

Which is very pretty I think. Anyway, over at Mathworld<\/a>, Eric Weisstein had written some Mathematica code to produce not only this variation of a pentaflake but also another one which was created by putting pentagons at the corners<\/strong> of the first one rather than the sides.\u00a0 Also, rather than using identical pentagons, this second variation used scaled pentagons for each iteration.\u00a0 The end result is shown below.<\/p>\n

\"pentaflake<\/p>\n

Looking at Eric’s code I discovered that it would be a trivial matter to wrap this up in a Manipulate function and produce an interactive version. This took about 30 seconds – the quickest Wolfram demonstration I had ever written. After submitting it (with due credit being given to Eric) I got an email back from the Wolfram Demonstration team saying ‘Why stop at just pentagons? Could you generalise it a bit before we publish it?’<\/p>\n

So I did and the result was named N-flakes<\/a> which is available for download on the Wolfram Demonstrations site. Along with pentaflakes, you can also play with hexaflakes<\/a>, quadraflakes and triflakes. One or two of these usually go by slightly different names – kudos for anyone who finds them.<\/p>\n

\"Quadraflake\"<\/p>\n","protected":false},"excerpt":{"rendered":"

Here in the UK we have had more snow than we have seen in over 20 years and as a country we are struggling with it to say the least.\u00a0 I have friends in places such as Finland who think that all this is rather funny…it takes nothing more than a bit of snow to […]<\/p>\n","protected":false},"author":3,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"jetpack_post_was_ever_published":false,"footnotes":"","jetpack_publicize_message":"","jetpack_publicize_feature_enabled":true,"jetpack_social_post_already_shared":false,"jetpack_social_options":{"image_generator_settings":{"template":"highway","default_image_id":0,"font":"","enabled":false},"version":2}},"categories":[46,6,18],"tags":[],"class_list":["post-626","post","type-post","status-publish","format-standard","hentry","category-fractals","category-general-math","category-wolfram-demonstrations"],"jetpack_publicize_connections":[],"jetpack_featured_media_url":"","jetpack_shortlink":"https:\/\/wp.me\/p3swhs-a6","jetpack_sharing_enabled":true,"_links":{"self":[{"href":"https:\/\/walkingrandomly.com\/index.php?rest_route=\/wp\/v2\/posts\/626","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/walkingrandomly.com\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/walkingrandomly.com\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/walkingrandomly.com\/index.php?rest_route=\/wp\/v2\/users\/3"}],"replies":[{"embeddable":true,"href":"https:\/\/walkingrandomly.com\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=626"}],"version-history":[{"count":21,"href":"https:\/\/walkingrandomly.com\/index.php?rest_route=\/wp\/v2\/posts\/626\/revisions"}],"predecessor-version":[{"id":3071,"href":"https:\/\/walkingrandomly.com\/index.php?rest_route=\/wp\/v2\/posts\/626\/revisions\/3071"}],"wp:attachment":[{"href":"https:\/\/walkingrandomly.com\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=626"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/walkingrandomly.com\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=626"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/walkingrandomly.com\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=626"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}