In 2007, we started BLOG Zentangle and began our enjoyable series of conversations within our Zentangle community.
In reading through these blog posts with their insightful comments, we decided to bring a few of them to your attention from time to time. It is easy, for me anyway, to sometimes think of old information as stale information. But these insights and conversations are anything BUT stale!
In 2018, Rick writes...
I want your advice.
This post contains math. I think it’s right, but I’m not sure.
To all you math wizards (or friends of math wizards), please tell me how I did.
It’s a good story even without the math. I’d love to read your thoughts.
Thank you!
About a year after our Zentangle adventure began, a friend and CEO of a regional corporation invited Maria and me to give a Zentangle workshop at his company’s annual meeting for executives and managers.
We gave our workshop after their lunch and before their meeting. About 40 people were there.
The tables were arranged in a horse-shoe fashion with people sitting on the outside. We stood at the open end with a large screen behind us to project what we would draw.
We handed out paper tiles, pens and pencils and introduced the Zentangle Method. As we guided people through the process, each person became immersed in their drawing. There was pin-drop silence in the room.
When we finished, every tile was unique even though you could see how each had followed the same guidance. Each person interpreted what we said through the lens of their individual expression.
Everyone placed their finished tiles on a card table in the middle of the horseshoe. Each tile was unique even though you could see how each had followed the same guidance. Each person interpreted what we said through the lens of their individual expression.
Together, the tiles formed a “mosaic” (which is why we call them “tiles”). As people added their tiles to the mosaic, you could hear their appreciation for other’s tiles and their often surprised appreciation for their own tiles as they viewed them in this larger context.
Coffee was served and people sat back down. We gathered our gear, said our goodbyes and they proceeded with their meeting.
We later learned that this annual meeting was the most productive and shortest in the company’s history. I like to think it was because of the card table holding their creations remained in front of everyone. The CEO’s tile was not distinguishable from the tile of the most recent hire. The mosaic that faced them was unique and could only exist because of each person’s unique tile.
After that workshop, I began thinking about a tile in a mosaic as a metaphor for an individual in a group. And I began thinking about the mosaic as a metaphor for a group.
Rearrange and rotate the tiles in different ways and you get a different mosaic.
To extend my metaphor, I wondered how many groups there might be if the same people sat in different seats with one of four different moods. So I made up a thought experiment for a group of 36 people.
Here it is.
Pretend each person has only two possible states:
- Tired or Rested and
- Hungry or Full
That means each person is either:
- Rested and Full, or
- Rested and Hungry, or
- Tired and Full, or
- Tired and Hungry
These options represent the four ways you can position a square tile in a mosaic.
Let’s also pretend there are six rows of six seats where the group sits.
The question for this thought experiment is:
“How many permutations can there be for a group of 36 people with two binary variables who can sit anywhere in six rows of six seats?”
Last week I decided to figure it out. When I did, I couldn’t believe the numbers! I’ll show you them at the end, but first let’s start with a smaller mosaic, one with only four square tiles. Before you scroll further, take a guess how many ways you can arrange this mosaic.
At first, I tried figure it by drawing pictures, but I soon realized that could take hours. So I searched online and ended up with this formulas for permutations of a certain number of tiles (or people) and a certain number of variables for each tile (or person):
(r^n x n!) / r
r = the number of possible rotations of a tile (3 for a 3Z and 4 for a square tile)
n = the number of tiles in the mosaic
(! represents the factorial of the number it follows. So 5! would be 5 x 4 x 3 x 2 x 1 = 120. I had to (re)learn that for this exercise.)
OK, let’s play with the four-tile mosaic.
(4^4 x 4!) / 4 = (256 x 24) / 4 = 6,144/4 = 1,536
There are 1,536 different ways you can arrange four tiles in a two by two mosaic!
Actually there are 6,144 ways you can arrange the mosaic from the perspective of any one side, but I decided to be conservative and assume that you can walk around the mosaic and view it from any side.
How many permutations do you think this three by three mosaic of nine tiles has?
Let’s do the math:
(4^9 x 9!) / 4 = (262,144 x 362,880) / 4 = 95,126,814,720/4 = 23,781,703,680
Yes, that’s over 23 billion (with a B) permutations!
Before we get to that meeting mosaic, let’s doing one more, a mosaic of 16 3Z tiles. Care to guess how many permutations of this mosaic?
Here we go . . .
(3^16 x 16!) / 3 =
(43,046,721 x 20,922,789,888,000) / 3 =
9.006574988504e20 / 3 = 3.002191662835e20 which looks like
300,219,166,283,500,000,000
Or, just “slightly” over 300 quintillion which is one billion times one billion times 300!
Okay, let’s solve my original question.
Ready?
Any guesses? (LOL)
Here we go:
(4^36 x 36!) / 4 =
(4.72236648287e21 x 3.719933267899e41) / 4 =
1.756688818284e63 / 4 =
4.39172204571e62 or in other numbers:
439,172,204,571,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000
or, a billion times a billion times a billion times a billion times a billion times a billion times 439,172,204.570951
Seriously?
Well, as I edited this I realized that if we’re talking about tiles which don’t change, but you can orient in four ways then, yes. But if we’re talking about individuals that can present themselves in four different combinations of tired/hungry, etc., then it’s that number times four.
A Zentangle mosaic is inspiring BECAUSE each tile is unique. What would be the fascination of a mosaic when each tile is the same as every other one?
What stunned me about this exercise is how MANY permutations exist for a mosaic of just a few tiles.
And when you appreciate how many more variables any individual has than the three or four variables of a tile . . . well . . .
The next time someone suggests you can presume an identity of a group based on the individuals, or the orientation of a single person because of their group, just get out 9 square tiles and your calculator and have some fun!
==
NOTE:
Another way you can approach this is to imagine four tiles and an empty 2 by 2 grid.
When you pick up the first tile you have four places in the mosaic you can place it and there are four rotations, in 90 degree increments to choose from. In other words, there are 16(4 x 4) ways to put that tile in the mosaic.
When you are ready to place the second tile in the mosaic, there are only three spaces left but you still have four ways you can orient that tile so you have 12 (3 x 4) ways to position that second tile.
With two remaining spaces for the third tile there are 8 options (2 x 4) and the last tile goes in the remaining space in one of four ways (1 x 4).
You will notice that we get the same result this way:
16 x 12 x 8 x 4 = 6,144
. . . which I divided by four for reasons stated above.
Perhaps a clarifying example will help:
4×3×2x1 is 4 factoral or 4! = 24 Mosaic of 4 tiles with Tile 1 in the upper left corner
4×3×2x1 Mosaic of 4 with Tile 2 in the upper left corner
Same for Tile 3
Same for Tile 4
Add this result (or multiple x4 for 4 tiles)
(4×3×2x1) x4 = 96 permutations (assuming all tiles as asymmetrical)
If a tile is symmetrical, turning it would not produce a new view of it. (Example of a 3Z tile that is symmetrical is on the bottom center of the photo with this blog. (Knightsbridge in diamonds). Turning the tile doesn’t change the mosaic.)
Comments regarding “the other side of the table”:
Interesting observation. It is correct to state that walking around the table gives a new “mosaic” or viewpoint; However, that new mosaic is also accounted for in this formula by shuffling the original position of the tile in the mosaic and rotating it a half turn. (Try that with just four tiles to see what I mean.)
Final note:
It appears to me that some of the formulas given are mixing combinations of 4, not permutations of 4. They are different. For example. a combination of 4 allows subsets to create a “mosaic” with only 3, 2 or even 1 tile.
Linda Dochter on
My math chops have REEEtired but I do have a few observations to contribute that I hope can be understood by everyone..
The formula to be computed is the number of permutations, not combinations. The formulas are different. Some of the comments do not appear to make a distinction. This exercise requires a permutation of 4. A combination would allow subsets such as a mosaic using only 3 out of 4 tiles or would allow one tile to be duplicated four times. I can’t really type the formula here but it would be 4×3×2x1 while Tile 1 was in position 1, then 4×3×2x1 while Tile 2 was in position 2, ect. That four factoral raised to the fourth power.
All of these answers assume every tile is asymmetrical (different view with each turn of the tile.) An example of a symmetrical tile is in the center of the bottom row of the photo shown (Knightsbridge over diamonds.) It is basically the same even from 3 views. the tile so just turning that tile does not change the mosaic.
Therefore each of the answers given is a theoretical maximum, not an actual maximum for a set of tiles picked at random.
The comment regarding the view from the opposite side of the table interests me. Good observation but it does not double the number of possible combinations. (Likewise viewing from either side of a square table.) Those views would be the same as the ones already counted from the original viewpoint by “shuffling the deck” and “turning the tiles” if you were to create the mosaic by hand. No up, no down, no left, no right.
I’m thinking that the answer for the maximum number of combinations (not permutations) is somewhat less than scaddy-eight-blue-jillion as suggested.
Linda Dochter on
Julia Davenport on
As a math atheist (see the cartoon strip “Calvin and Hobbs” for the reference), numbers make my brain hurt. The higher the math, the more my brain implodes. I was only good at calculating parabola and hyperbola because they created art, even if they were only arcs.
Mosaics, however, make my heart soar with inspiration! All the ones I witnessed last week at CZT38 were so amazing and interconnected into one beautiful whole of collaboration. It was a joy to no longer be alone with my tangling but instead, be with a group of diverse people who cherish it as much as I do in my life.
Debbie Smith CZT38 on
I totally admire people who are “numbers oriented”, like my husband for one, but I have always been a “words oriented” and “visual leaning” person,so to read through your amazing reasoning, I believe it all, but it darn near made my eyes bleed and I truly need a cup of coffee and a good lie down now🤪😅
Sue Zanker on
Ginger White CZT34 on
Wow! This blog was just plain confusing. I really don’t care how many times you can do the math, I just love to draw the tangles! Sure, I’ve flipped my tiles to different orientations, but never thought about how many times a person can do that. I just enjoy the many aspects and views of the tiles. Using math in this just makes it wayyyyyyyyy too confusing and takes the fun out of looking at the tiles. Sorry, Rick….no math please…just enjoy the tiles and the creativity.
Dianne Riva Cambrin on
Jessica Dykes (aka, Jake) on
Kathy Y. on
Jan ~Sailandbejoyful~ on
Jeanne Hertzel on
Victoria Smith on
@Lisa Locke
When you assemble a mosaic of square tiles on a table, is it a different mosaic if you look at it from the other side of the table?
I decided the answer should be, “no” for both practicality and to be as conservative as possible.
If you could live long enough to arrange the mosaic in its full set of permutations, and if you had an eidetic memory, you would notice that one quarter (in the case of square tiles) of the permutations reappeared in rotations of 90, 180, and 270 degrees . . . in other words, as if you viewed it from different sides of a square table.
Therefore I divided by r.
RIck Roberts on
LadyD on
I came up with 256 permutations for four tiles. Each tile can be in one of four positions. So, that is 4 tiles x 4 positions = 16 permutations. Each tile can be turned 4 ways. So, that is Side 1 + Side 2 + Side 3 + Side 4 for a total of four possible directions each tile can face. So, it would be 4 positions x 4 sides = 16 possible permutations again. And, 16 possible positions x 16 possible directions = 256 possible permutations for four tiles with four sides. I dropped this math course though and changed to trigonometry. It was easier :) I hope you find a math genius to solve this for us.
Pixel Ghost on
I think your math needs revisiting. 4 tiles would be 4!, which is 4×3×2x1=12.
The combinations would be:
1 2 1 2 1 3 1 3
3 4 4 3 2 4 4 2
1 4 1 4
2 3 3 2
That’s six combinations with tile one in upper left. Four tiles x 6 is 24.
Nine tiles would be:
9×8×7x6x5×4×3x2x1 = 362,880. Hope this helps.
I love Zentangle and love math.
Gail Minichiello on
I quickly got (r^n x n!), but I don’t understand why you’re dividing by r.
Lisa Louque on
Heavens To Murgatroyd! So when are you taking this to the United Nations?
Jean Mackie on
Thanks for all the comments! The math hasn’t been independently verified yet. But the newsletter that’s coming out will link to this blog. I’m looking forward to either verification OR correction.
Rick Roberts on
Did the mathematics ever get confirmed?
Laura Story on
Rita miller on
I never thought about all the possible variations of a mosaic like this before…
Jackie on
Tanja Francke on
Wow, I guess I have been out if school too long, because I get 40 for the original problem of 4 tiles on a 2×2 grid. Back to tangling…
Dana "Jonesy" Jones on
There is a character in the book The Phantom Tollbooth called the “Mathemagician.” That’s you, Rick! Thank you for sharing the marvelous infinite number of possibilities that Zentangle offers to all of us!
Jane Rhea on