Talk:Pascal's Triangle

Summer 2010

Work on the layout of images

Image 1, and the images in the Fibonacci section, seem awkwardly placed, to me at least. Fiddle with them a bit. Part of the problem with Image 1 may just be that it's unnecessarily large (Abram, 7/15)

Iris(7/19) fixed this
Looks good. (Abram, 7/22)

Consider getting rid of Pascal's representation of the triangle completely

Having a small version, at least, of Image 1, is kind of interesting for historical reasons. I wonder if the proof that uses Pascal's representation could be altered to use the modern representation, just because it's easier to follow. (Abram, 7/15)

Iris(7/19) I agree, I got rid of it
Looks good. (Abram, 7/22)

Make the "combinations" interpretation of Pascal's Triangle more central

We may want to *define* Pascal's Triangle this way, and describe the Pascal's Relations rule as a *property*, if I'm right that this is the standard view today. In any case, we almost certainly want to put this definition near the beginning of the page (see threads about large-scale restructuring of patterns and large-scale restructuring of applications, below).(Abram, 7/22)

Large-scale restructuring of patterns

Right now, it seems like everything that has a proof is in the more mathematical section, while everything that doesn't have a proof is not. This organization probably doesn't make sense, because there's no intrinsic reason why the Fibonacci sequence pattern is less mathematical than the hockey stick pattern and if we do eventually include proofs for these properties, the whole page will be the more mathematical explanation. This is silly, because all the patterns, and even some of the applications, are perfectly understandable for any reader. It's just the proofs that are harder. Here are a couple ideas we came up with:

• Don't have *any* section called the More Mathematical Explanation (except maybe the Binomial Coefficents). Just hide all the proofs and make sure to tell the reader that there's no harm in skipping over these proofs. One good reason for doing this is that many of these patterns can be proven without any equations by using the "combinations" interpretation of Pascal's Triangle, so those proofs can be made available to every reader.
• Have the basic description consist of a description of all the patterns, and perhaps eventually a non-equation based proof, while all the equation-based proofs are in the more mathematical explanation.

(Abram, 7/22)

Large-scale restructuring of applications

It seems like we definitely want the "combinations" interpretation of Pascal's Triangle put near the beginning of the page because it eventually makes way for non-equation-based proofs of just about everything. It's also possible that the heads/tails application should be put before the patterns. It seems that more readers are interested in applications than are interested in patterns (because they often see the patterns as kind of cheap carnival tricks).

Finally, the combinations interpretation should definitely definitely come before the heads and tails section. (Abram, 7/22)

Think of ways of getting interesting stuff into the basic description or a why it's interesting section

Possibilities include: moving some of the patterns to the why it's interesting section (the math isn't very complicated); giving an overview in the basic description of the types of things the triangle is useful for or why Pascal invented it; anything else you can think of. (Abram, 7/15)

Hmm, actually, this issue has probably been replaced by the issues above. (Abram, 7/22)

Rewrite the coin-tossing section

I know you were trying to work with what was already there, but most of those sentences are kind of confusing. You can decide whether it's easier to fix it sentence-by-sentence or to just rewrite the whole section. (Abram, 7/15)

Summer 2009

Lizah 7/14

Hey Abram,

I tried changing the wording. Is it clearer now?

Abram 7/14

Hey Lizah,

Pascal's triangle can be used to determine the combinations of heads and tails we can have depending on the number of tosses. From the possible outcomes, we can calculate the probability of any combination.
For example, if we toss a coin twice, we could have the any of the following combinations: HH once, TH twice and TT once, thus the possible outcomes would be in the order 1 2 1.This is also the same as the second row of Pascal's triangle. In general, if we toss a coin $x$ times, the combination of possible outcomes would be the horizontal entries in the $x$th row of the Pascal's triangle.

What is a "possible outcome", what is a "combination", and what is a "combination of possible outcomes"? It seems like a "combination" refers to a specific number of heads and tails, and a "possible outcome" refers to a specific sequence of coin-flips, but it might help to define those terms in the page, and I'm still confused about what a "combination of possible outcomes is".

In this last sentence, it may actually clarify things to say what the hth entry of the xth row represents, rather than to say what the xth row as a whole represents.

Abram 7/10

Looks nice. Really good changes. Last fixes:

The hockey stick picture in the basic description is too small.

Your entries for 11^5 and 11^6 in the Magic 11's table aren't correct, because the pattern works just a bit differently than you think it does. It's easier for me to explain the problem out loud.

When you describe the triangular numbers, etc, put a link to the animation.

//the combination of possible outcomes would be the horizontal entries in the xth row of the Pascal's triangle.// This section is great, but what do you mean "the combination of possible outcomes"? I think you might be saying something like: "If you flip a coin x times, the hth entry tells you the relative likelihood that exactly h of those flips will come up heads (and the rest will come up tails)."

Also, you seem to have some repeated entries in the 3rd row of that table.

Anna 7/10

I'd edit this sentence

"If you add the numbers in each row horizontally, then look at the pattern formed by the resultant numbers, you will notice that th numbers double each time, but all are in the power of 2."

So that it reads "If you add the numbers in each row horizontally, then look at the pattern formed by the resultant numbers, you will notice that the numbers double each time and are the powers of 2."

Abram 7/9

This is a rich and well-constructed page. I have just a few suggestions before I would declare the page ready for public.

First, I would second all of Chris's comments below.

I would also add a few things.

Properties section: Add a mouse over to the words "triangular number", "tetrahedral number", and "d-simplex numbers" giving rough definitions of the words. These words are all easily understood (even d-simplex can be roughly understood) without any heavy mathematical terminology, but I wouldn't just assume that readers know any of those words if you don't define them.

Use of the word "tutorial": Rename the section that is called "Animation Demonstrating Patterns" as "Tutorial animation", or replace every use of the word "tutorial" with "animation demonstrating patterns". Also, put an anchor in front of the animation with a link to that anchor every time you use the word "tutorial" or "animation demonstrating patterns" (whichever one you choose).

Fibonacci numbers: Hide the definition of the Fibonacci numbers in a mouse over.

Chris 7/9

I love this topic. It is so rich and has so many applications.

Properties: “The next d-diagonal contains the next higher dimensional “d-simplex” numbers.” I notice that simplex is a link to wikipedia, but it’s awkward to have a sentence that depends upon a word so complicated to understand that it needs such a link.

Magic 11’s: The single numbers for Row 5 and Row 6 are not correct. What happens is that the double digit numbers overlap. Row 5 becomes 161,051, not 15,101,051.

Hockey Stick Pattern, last sentence: What should we refer to?

I see your reference to combinatorics, I would love to see something about how Pascal’s Triangle is related to flipping coins. The number of times a coin is flipped is the row, and the number of possible outcomes for each combination corresponds to the place on that row. It’s a very basic and elegant application of (or hook into) Pascal’s Triangle.

Abram 6/25

Lizah,

I like the approach you've taken to this page, which seems to be presenting a catalog of some of the most interesting properties of the triangle. Many of the patterns in the triangle are quite difficult to describe using words, and in many places, you have done a great job using slightly fuzzy language to get your point across in a surprisingly clear way.

Small but important changes:

• The image description is obviously pretty boring. Can you tell us right at the beginning that Pascal's triangle is a triangular arrangement of specific numbers that has lots of interesting patterns, and maybe even allude to one of those patterns right at the beginning.
• You write, "Pascal's triangle is a gigantic pattern in itself," which is not true. Pascal's triangle is a triangular arrangement of specific numbers. It is true that the method you describe for generating the entries uses a simple rule, but Pascal's triangle is not itself the rule.

Main big change:

I'm actually fairly confused by the basic description.

• The easiest way to address this is to have an image or animation that shows the entries of Pascal's triangle being constructed. The one on the wikipedia page isn't bad, but it's also not great. The alternative to finding or creating an image/animation is to expand your wording a lot. I say get the Drexel folks on it, or search for images. Then reword your explanation to make specific references to the animation.
• Either way, your use of "consistent pattern" in the first sentence is confusing. It a)uses the term "pattern" differently than you use the word for the entire rest of the page, and b)makes it sound like it can follow any pattern, so long as it's consistent. How about instead of "follows a consistent pattern" you say, "is created using a specific method".
• "This pattern is just one of the many patterns within the triangle" is similarly problematic because you present this pattern as a construction method. It's true that the fact that each entry is the sum of the two "directly" above it, is a pattern, but it's weird to describe your working *definition* as a "pattern".

Second big change:

Move some of the patterns in the more mathematical section to the basic description.

• The hockey stick pattern can basically be moved as is, and the Sierpinski triangle pattern, with just a bit more explanation, can also be moved.

Basic description: How about a couple small examples to show the pattern?

We can continue doing this endlessly, for this reason, Pascal's triangle goes to infinity. [You don't really need the rest of the sentence.]

This pattern [which pattern?] is just one of the many patterns within the triangle "as we shall see" [or something].

"70 for example would be identified as row 8, place 4" Some smaller examples with pictures, please.

Properties: The next set of numbers inwards after the natural numbers are triangular numbers 1,3,6,10,15... [define triangular number, tetrahedral number, etc?] Give a ref to proof of last property.

Magic 11's: right before table don't you want "nth"? (superscript h didn't make it).

Your properties are delightful. I'd like the names to stay even when the rest of the stuff is hidden (and lots of "shows"). Good that you don't do too many but leave some in red for others to do!

"This is summarized in the binomial theorem" but I think you need a little more info.

Very good, Lizah. Particularly good at choosing a good amount to show and a good amount to leave out.

I can has text? -Emily G.

Hey Lizah, would you object to me borrowing some of your text from this wiki and copying it straight into the animation to describe some or all of the patterns? This could perhaps give you the chance to remove some of the text from the page, reducing clutter, or the chance to expand upon these patterns even further. It's up to you.

P.s.- I just scrolled down to submit this comment, and read the disclaimer which basically says that your work is free to use, but I still think I should ask.

Hey Emily,