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Permutation: The act of changing the arrangement of a given number of elements.
One font, two different brick combinations.
Picking any two bricks from the 169 available gives a total possible combinations of 14196 (169C2) different fonts. Counting a certain kinds of bricks as one--all four 45degree, for instance--gives 36 unique bricks, resulting in 630 (36C2) unique combinations or fonts.
In this font, if the bricks are swapped with each other, the result will be a different font. Hence order of the bricks matter. In which case, nCr (combinations) is not the right choice. What's needed is nPr (permutations). 169P2 gives 28392 permutations and a 36P2 gives 1260 permutations.
So, at a minimum, 1260 fonts are possible with the current implementation of FontStruct, with just this particular layout of bricks.
Staggering.
6 Comments
This has to do with the mathematics of permutations and combinations, and is quite useful in probability calculations.
Note: some sections of the following are paraphrased and/or out-right lifted from their respective Wiki pages -- it's been way too long (almost 18 years) since I took a Probability course in college...
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A permutation is an arrangement of items in a particular order. The standard formula to count all possible item permutations is:
Permutation = nPr = n! (read: n factorial, where "!" is the mathematical symbol for factorial). In expanded form:
n! = n×(n-1)×(n-2)×...×1 for all non-negative integers.
In a factorial, every number starting with n and all numbers down to 1 are multiplied together.
So, as an example, 5! = 5×4×3×2×1 = 120
For the special cases of 1 & 0, 1! = 1 and 0! = 1.
When not counting all of the items, the formula is nPk = n!÷(n-k)!, where n = number of items total, and k = number of subsequent items picked.
For example, how many different yet unique ways can a single poker hand (5 cards) be dealt from a standard deck of 52?
n = 52; k = 5
52P5 = 52! ÷ (52-5)!
= 52! ÷ (47)!
= 52! ÷ 47!
= 52×51×50×49×48×47! ÷ 47!
= 52×51×50×49×48
= 311,875,200 unique 5-card poker hands can be dealt.
When all items are picked from the set, k = n and thus the formula becomes:
n = n, k = n
nPk = n!÷(n-k)!
= n!÷(n-n)!
= n!÷0!
= n!÷1
= n!, which was initially stated above for nPr.
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A combination is a way of selecting several things out of a larger group, where (unlike permutations) order does not matter. The formula for Combinations is n!÷(k!×(n-k)!), where n is the total number of items, and k is the number of items chosen to be in the subset.
Combination = nCk = n!÷(k!×(n-k)!)
For example, given three fruit, say an apple, orange and pear, there are three combinations of two that can be drawn from this set: an apple and a pear; an apple and an orange; or a pear and an orange.
n = 3, k = 2
3C2 = 3! ÷ (2!×1!)
= 6 ÷ (2×1)
= 3
The Permuation would be as follows:
n = 3, k = 2
3P2 = 3! ÷ (3-2)!
= 6 ÷ (1)
= 6
And here are those permuations:
--Apple first + Orange second
--Apple first + Pear second
--Orange first + Apple second
--Orange first + Pear second
--Pear first + Apple second
--Pear first + Orange second
Note that Permutations care which fruit is picked first and then which is picked second, whereas Combinations care only about what is in the set and not their "picking order".
So, how about a single poker hand where the order of the cards dealt does not matter?
n = 52, k = 5
52C5 = 52!÷(5!×(52-5)!)
= 52!÷(5!×47!)
= 52×51×50×49×48×47! ÷ (5!×47!)
= 52×51×50×49×48 ÷ (5×4×3×2×1)
= 311,875,200 ÷ 120
= 2,598,960 single poker hands can be dealt with the 5 cards showing up in any order.
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So, from Thalamic's description:
169C2 = 169! ÷ (2!×167!)
= 169×168×167! ÷ (2!×167!)
= 169×168 ÷ 2
= 14,196
36C2 = 369! ÷ (2!×34!)
= 36×35×34! ÷ (2!×34!)
= 36×35 ÷ 2
= 630
169P2 = 169! ÷ (167!)
= 169×168×167! ÷ (167!)
= 169×168
= 28,392
36P2 = 36! ÷ (34!)
= 36×35×34! ÷ (34!)
= 36×35
= 1,260
**********
Hope this helps!
@Frodo7: Swapping is when one brick in MyBricks is replaced with another (from either AllBricks or from within MyBricks). On any brick in AllBricks (even ones already used), LeftClick and hold the mouse button and drag the mouse to an existing brick in MyBricks. For instance, if you had the square brick in MyBricks, you can replace it with, say, the star brick. Wherever the square brick is used in the fontstruction you are working on, it will get replaced with the star brick.
The fs Permutation series is built using this technique. A basic font was created using only two bricks — one for the foreground (glyphs) and one for the background. The fs was cloned and each of the brick was swapped with some other brick.
As for permutation and combination: Suppose you are building a fruit salad containing apples, peaches and grapes. It doesn't matter which fruit you chop and add to the bowl first. Therefore, the order doesn't matter. This is an example of combination. Now suppose you are building a tower in Lego® bricks and you have a limited number to do it with. Which brick you use where will make a difference in how the tower turns out. You can undo the tower and rebuild it in some other placement of the exact same bricks used to build the first tower. This will result in a new style — or permutation — of the tower. Therefore the order mattered.
Taking this fs series, there are only two brick in use, the square and the small dot. Suppose, the foreground part of the glyph is created using the full square brick and the background with the small dot. It will result in a prominent letter over a dotted background. If we reverse the brick usage, with the letter in small dot and the background in the full square brick, the result will be a negative of the first experiment. Therefore, as far as fonts are concerned, the order of brick placement makes a difference.
As you can see from Goatmeal's explanations, with the same limited number of objects to start with, permutations give a much larger number of possibilities as opposed to combinations. At the time when this fs series was initially created, there were a 28,392 permutations possible if each brick was used individually and 1,260 permutations possible if each brick was used in sets (e.g.: all 4 triangle bricks to be counted as 1).
This was just and experiment. It gave rise to startling results.
I enabled Clone. Test it out. It's fun and informative.
HTH.
It is wonderful to see typeface design at the intersection of art and mathematics.
I hope, I wasn't the only one ignorant to these details, so your explanations could benefit others too shy to ask questions. Thanks again.
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