This is what Theoni Pappas writes on page 125 of her Magic of Mathematics:
"Of the three (hexagon, square, and equilateral triangle) the hexagon encompasses the largest area with the minimum amount of material. Another feature of the regular hexagon is that it has 6 lines of symmetry, thereby allowing its shape to rotate numerous times without altering."
We've already seen the first property which, of course, explains why honeycombs are hexagonal. So let's look more at the second -- the symmetry of the regular hexagon.
Thinking back to Common Core transformations, we see how Pappas mentions the regular hexagon's lines of symmetry allows its shape to rotate. But wait a minute -- "lines of symmetry" refers to reflection symmetry, yet Pappas then connects this to rotations. What gives?
Well, it's been some time since I wrote about this. But recall the definition of rotation -- the composite of two reflections in intersecting lines. Thus, a figure with at least two intersecting lines of symmetry automatically has rotational symmetry. The regular hexagon has six symmetry lines all intersecting at the center. Therefore, it does have plenty of rotational symmetry.
We can determine the magnitude of the rotation as follows -- the six symmetry lines form twelve congruent angles at the center, so each angle is 360/12 = 30 degrees. The magnitude of the rotation is double the angle of intersection, so the rotation is 60 degrees. Of course, for regular polygons, we can just divide 360/6 = 60 degrees to find the angle of rotation as well.
This is what I wrote last year about today's activity:
Today I finally post the vector activity that I've been planning this week.
There is also a page to cut out with 36 given vectors -- since as I mentioned earlier, I don't want the students to choose the vectors. Of course, cutting out the vectors is time consuming -- even if the teacher does it before the class -- and those tiny slips of paper are easily lost. Then teachers will have to cut them out several times throughout the day.
Another way to have the students choose vectors would be to have something larger represent the vectors, such as playing cards. The playing cards can be converted to vectors, as follows:
For the horizontal component:
Ace through 10 -- valued 1 through 10
Jack -- valued -2
Queen -- valued -1
King -- valued 0
For the vertical component, use the suit:
Clubs -- valued -1
Diamonds -- valued 0
Hearts -- valued 1
Spades -- valued 2
Eight of Diamonds -- (8, 0)
Ace of Clubs -- (1, -1)
Jack of Hearts -- (-2, 1)
Because they are larger, playing cards are less likely to be lost than the little slips of paper that I provide for this activity. But there are problems with using playing cards. First of all, the conversion from playing card to vector is another step that the first partner can get wrong -- and once again this may frustrate the second partner. (I've heard that some people don't even know the difference between clubs and spades!) Furthermore, vectors with large components, such as (8, 0) for the eight of diamonds above, become (30, -2) after performing the steps in Task Three -- and then they are asked to graph that vector (30, -2) in Task Four. So I leave it up to individual teachers whether to use playing cards or the slips of paper that I provide.