tag:blogger.com,1999:blog-1920088135580776574.post6871771812518938813..comments2013-10-28T08:30:20.487-07:00Comments on Ask Doctor Math: Wholly Whexagons!drmathhttp://www.blogger.com/profile/17936175968300765200noreply@blogger.comBlogger10125tag:blogger.com,1999:blog-1920088135580776574.post-27548778102280432892012-11-17T00:45:18.815-08:002012-11-17T00:45:18.815-08:00Thank you for this awesome post. I also love Hexag...Thank you for this awesome post. I also love Hexagons. They are very common in board games. I'm currently working on a android video game that takes a lot of inspiration from D&D and board games. I'm going to use hexagons to lay out tiles for the battles. The beauty is that instead of squares where you have 4 places to move/attack to/from you instead have 6 increasing the fun by 50%.Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-1920088135580776574.post-90016723661723738362009-03-05T19:28:00.000-08:002009-03-05T19:28:00.000-08:00Nice, Mike! Now why don't we have that same "geom...Nice, Mike! Now why don't we have that same "geometrical forethought"?drmathhttps://www.blogger.com/profile/17936175968300765200noreply@blogger.comtag:blogger.com,1999:blog-1920088135580776574.post-81197914488466856672009-03-05T08:02:00.000-08:002009-03-05T08:02:00.000-08:00Dr Mfrom a pre-blog era...Pappus of Alexandria. ca...Dr M<BR/>from a pre-blog era...<BR/><BR/>Pappus of Alexandria. ca 290-350. Greek geometer.<BR/><BR/>"Bees. . . by virtue of a certain geometrical forethought . . . know that the hexagon is greater than the square and the triangle and will hold more honey for the same expenditure of material in constructing each".<BR/><BR/>mikeAnonymousnoreply@blogger.comtag:blogger.com,1999:blog-1920088135580776574.post-64659986021556809292009-03-04T16:34:00.000-08:002009-03-04T16:34:00.000-08:00Hmmm... I'm not an expert in this sort of thing, b...Hmmm... I'm not an expert in this sort of thing, but one idea I have is to exploit the fact that a hexagon is composed of six equilateral triangles, for which there are perhaps better construction techniques.<BR/><BR/>For example, starting with two points a fixed distance <I>d</I> apart, you can use a compass to draw a circle of radius <I>d</I> at each point. Taking (either) point of intersection of the two circles gives you an equilateral triangle with side length <I>d</I>. No need for a protractor. If you repeat the process six times, you should get a nice regular hexagon.<BR/><BR/>I'm interested to hear other people's thoughts, though. Let me know how it works out for you!drmathhttps://www.blogger.com/profile/17936175968300765200noreply@blogger.comtag:blogger.com,1999:blog-1920088135580776574.post-30855540355941210972009-03-04T12:16:00.000-08:002009-03-04T12:16:00.000-08:00Wonderful! I am building a brick patio with a cent...Wonderful! I am building a brick patio with a central design that is based on a hexagon. I'd love to hear some tips on geometric constructions that would be easy to lay out physically.<BR/><BR/>To lay out my hexagon-based shape on paper I draw a large circle and one diameter line. Using a protractor from the center off the diameter I draw two additional diameters at 60 degrees off.<BR/><BR/>Next I draw in chords where the diameters intersect the circle to form the hexagon.<BR/><BR/>Since the result of my shape is a flower I next form circles whose center is the midpoint of each chord (hexagon side) that intersect at the corners of the hexagon.<BR/><BR/>I potentially see some trouble with these last steps in the real world, where I'd be using less accurate tools and methods, but would like to end up as perfect a result as possible.<BR/><BR/>I'd love to hear more from you about how people used to lay out complex but regular geometries for floors and architecture using simple tools.John Thttps://www.blogger.com/profile/00432574977650511638noreply@blogger.comtag:blogger.com,1999:blog-1920088135580776574.post-74606799731889636482009-03-04T08:11:00.000-08:002009-03-04T08:11:00.000-08:00Whoa, I never noticed that. I'll have to check it...Whoa, I never noticed that. I'll have to check it out.DUhttps://www.blogger.com/profile/15427884103652875815noreply@blogger.comtag:blogger.com,1999:blog-1920088135580776574.post-43710592135463712862009-03-04T08:09:00.000-08:002009-03-04T08:09:00.000-08:00Right, good catch, DU. I should have been more cl...Right, good catch, DU. I should have been more clear about that.<BR/><BR/>As a side note, I believe pentagonal nuts/bolts do get used sometimes--for example, on <A HREF="http://www.firehydrantcaps.com/products" REL="nofollow">fire hydrants</A>--exactly <I>because</I> you can't open them with a standard wrench.drmathhttps://www.blogger.com/profile/17936175968300765200noreply@blogger.comtag:blogger.com,1999:blog-1920088135580776574.post-42816281254292711092009-03-04T06:46:00.000-08:002009-03-04T06:46:00.000-08:00Squares, octagons, and other n-gons have the same ...<I>Squares, octagons, and other n-gons have the same property [of having parallel opposite sides]...</I><BR/><BR/>But only for even n. You wouldn't want a triangular (n=3) or pentagonal (n=5) bolt.DUhttps://www.blogger.com/profile/15427884103652875815noreply@blogger.comtag:blogger.com,1999:blog-1920088135580776574.post-60306288034349914742009-03-03T22:38:00.000-08:002009-03-03T22:38:00.000-08:00Good points, Daniel! Of course, where things real...Good points, Daniel! Of course, where things really heat up is in three dimensions, where once again the hexagon makes an appearance in the problem of optimally <A HREF="http://en.wikipedia.org/wiki/Close-packing" REL="nofollow">packing spheres</A>. Is there no limit to the awesomeness of the hexagon?drmathhttps://www.blogger.com/profile/17936175968300765200noreply@blogger.comtag:blogger.com,1999:blog-1920088135580776574.post-21504941421689165972009-03-03T22:29:00.000-08:002009-03-03T22:29:00.000-08:00Don't forget to mention that hexagonal packing is ...Don't forget to mention that hexagonal packing is exceptionally common at the atomic level. It is the densest form of packing (on a plane). Picture those oranges as atoms, and it is obvious.<BR/><BR/>Another interesting, and related, property is that a hexagon is THE shape that is formed when you circumscribe a circle with other like-sized circles. I.E., if you take one circle, you can surround it perfectly with 6 circles on the outside. No other number of circles will surround it tightly.Daniel Grossbergnoreply@blogger.com