6
$\begingroup$

The following image has been in the news recently:

enter image description here

My understanding is that these are all the known (to-date) tilings of the plane using convex pentagons.

Can someone explain to me why the following are not in the list?

$\endgroup$
10
$\begingroup$

I don't have enough reputation here to comment, but I wanted to clarify Deusovi's answer.

The first tiling (Floret) is type 5, shown in the upper right of the diagram. The next tiling (Cairo) is type 4, shown in the middle of the diagram. The last tiling (Prismatic) is type 1, shown on the upper left of the diagram.

The first 1-13 types are sets of equations that each define an infinite number of valid pentagons. For the exact equations you may refer to http://www.mathpuzzle.com/tilepent.html.

Type 14 and Type 15 describe the pentagon angles exactly, allowing only for changes in scale.

$\endgroup$
  • 1
    $\begingroup$ Thanks, although the answers posted now prompt the question: what exactly is the difference between, e.g., patterns 4 (dark blue) and 8 (medium gray)? $\endgroup$ – kjo Aug 15 '15 at 12:28
  • $\begingroup$ The symmetries are different between 4 and 8. For example, look at the points where four pentagons meet. In 4, it's the same vertex of all four pentagons, and the meeting point is a center of rotational symmetry for the entire tessellation. In 8, different vertices meet at those points, and no such symmetry exists. $\endgroup$ – Jonathan Lidbeck Apr 3 '18 at 0:03
10
$\begingroup$

They are - top right, middle, and top left. It's just that the ones in the diagrams use more "general" forms by not assuming any congruent sides or angles

$\endgroup$
  • $\begingroup$ If they are shown not assuming congruent sides or angles, what is the difference between: Type 1 and 6. Also between Type 4 and 8 ? You can convert them to selves just by slightly changing the angles/sides. I don't get it. $\endgroup$ – Mikhail V Jul 16 '16 at 12:43

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.