Talk:Three Dimensional Pythagorean Tree

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Contents

Checklist for Writing Pages

Added by Greg Brown.

Messages to the Future

  • Might be worth talking about total surface area and volume after multiple iterations.
  • Could also talk about non-isosceles cases.

Reference and Footnotes

  • The only reference is for the main image, which is cited.

Good Writing

Context

  • It seems like the main attraction is that this is a fairly basic fractal, which produces an interesting image.
  • The Why It's Interesting section seems a little bare to me? It mainly just restates stuff about fractals and the Pythagorean theorem, but those are really the core ideas. Is there more to say?

Quality of Prose and Page Structure

  • I have worked on making the structure of the page more "deliberate," ie. I've made the Basic Description more clearly a description of what the Pythagorean tree is (I also split the making of the Pythagorean Tree into its own section).
  • I've related the table of values in the More Mathematical Section to the sections that follow. I've tried to make it clear which relationships the page will be investigating.
  • I am a little concerned with the way I define fractal - is it introducing unnecessary vocabulary too early?
  • There is not really any difficult math on this page. There is basically no math outside of the More Mathematical Section. The "toughest" parts are the proofs, which make use of the Pythagorean theorem.
  • The Making of the Pythagorean Tree section seems a bit out of place to me. I realize that a large part of what the students were doing was attempting to replicate the tree, but I suppose I don't find the process of doing so all that relevant, ie. the benefits and limitations of SketchUp, and the resulting process required to make the tree don't seem to enhance a reader's understanding. How do we include this section without leaving readers wondering why it's there?

Integration of Images and Text

  • The images are all referred to in-text as appropriate.
  • Some of the images end up looking kind of bulky because in some sections (particularly the Making the Pythagorean Tree section) the images take up a lot more space than the text, so the page feels sparse (this might be a result of my browser being widescreen). I can't really table the image or right/left-align them since they would either be too wide or would extend below into the More Mathematical Section.
  • I could go through and make the images all thumbnails w/ descriptions and anchors for in-text reference. It seems more appropriate to do this after we decide if anything need be added/removed/moved.

Connection to Other Mathematical Topics

  • This is a very basic topic, so I am not sure what I would link it to.
  • I did not see a page that was a general introduction to fractals, although such a link might help better than what I am currently doing - putting the definition of fractal in a balloon.

Examples, Calculations, Applications, Proofs

  • The image in the More Mathematical Section gives preliminary calculations for the relationships that will then be proved. Otherwise the page does not really have examples, but that seems like all that is necessary. Perhaps some of these relationships should be spelled out more in-text?
  • The page originally gave numerical examples of some of the proofs before proving them, but these followed the pattern of just modifying the Pythagorean theorem rather than, strictly speaking, proving the relationship.

Mathematical Accuracy and Precision of Language

  • I've refined a lot of the language throughout the page.
  • In particular I have redone a lot of the proofs. The old proofs mainly assumed the Pythagorean theorem and then modified it to get some result. While the idea was right, they weren't really in proper format. Now the proofs identify that the relationships we are looking for involve finding some constant that relates the base cube to its successive cubes. Then we verify that constant.
  • I try to reinforce the ideas behind this (and how they are unified in the Pythagorean theorem) in the Wrapping Up section.

Layout

  • I've added a lot of balloons and bold-faced definitions to facilitate the beginning of the page.
  • I think the text is broken up into reasonably sized paragraphs.
  • As mentioned above, some of the spacing and formatting gets funky in the Making the Pythagorean Tree section, mainly because there are a lot of images and relatively little text.
  • The More Mathematical Section contains most of the math and all of the proofs. There is no other "layer" of hiding things, as it would not be necessary; the math does not get substantially more difficult at any point, and the proofs are short and straightforward.
  • The page seems to look fine (even better, in certain ways) in smaller window sizes, although a couple images are rather large, so the window size cannot be too small.

Other

Hey guys! This is Greg from Swarthmore. You have done a lot of good work on this page so far. The goal for this summer is to polish it up further so that it can be published. I will make some suggestions here and you guys can work into the page what you'd like. I know it's summer so if you would like to leave the page as is, that's fine, just let me know and I will continue working on it. I am going to make suggestions as to the content, structure, proof strategy, and possible future directions of the page.

Content
  • You open by saying: "A 3-D Pythagorean Tree is a geometric figure that uses cubes connected to the sides of an isosceles right triangular prism to create a shape not unlike that of a tree."
    • It might help to fit the word "fractal" somewhere in this first sentence (or the term "self-similar").
    • You want to be clear that the tree is called Pythagorean because in each iteration, the side lengths of the cubes adhere to the Pythagorean theorem. This is implicit in the mention of a right triangle, but it is possible to make it clearer.
    • It is also relevant that the triangles formed are not necessarily isosceles, but for the purpose of the page you are addressing the case in which they are, so you might say something like, "A right triangle is required, although on this page we will consider the case that the triangle is also isosceles."
    • You may also want to distinguish the 3D case from the 2D case, although that distinction may fit better in the Basic Description.
  • Right now the Basic Description describes how to make a 3D Pythagorean tree in SketchUp. Here, you want first and foremost to be clear (but not too technical) about what the 3D Pythagorean tree is. That could include the fact that it's a fractal, what a fractal is, how it differs from the 2D version. It might be worth describing the 2D case in detail and then describing the 3D case as the extrusion of the 2D case into cubes (since that really is the primary difference).
    • It might also be worth specifying that the triangles are not part of the tree; they are "holes" and are important because they structure the relationships between the side lengths of the cubes.
  • You begin the More Mathematical Explanation by saying: "The 3D Pythagorean Tree is a manipulation of the Pythagorean Theorem of a^2+b^2=c^2 into the third dimension."
    • It would be more accurate to say that the 3D Pythagorean tree is a manipulation of the 2D Pythagorean tree into the third dimension, by turning each square into a cube with depth equal to its side lengths. The side lengths then still adhere to the Pythagorean Theorem. Because it is composed of cubes, we can then examine relationships among the cubes like surface area and volume (as you do in the More Mathematical Explanation), which don't apply in the second dimension.
  • It might help to clarify what the purpose of the table towards the beginning of the More Mathematical Section is.
    • In mathematics, we can calculate things with numbers or we can more generally find relationships using variables. What you seem to be doing with this tree is "exploring" the relationships of the 3D Pythagorean tree by doing some preliminary calculations, and the results then guide the proofs you do later in the More Mathematical Section. It might help to explicitly state that.
    • You might also conclude this "intro" to the More Mathematical Section by saying something like, "Such and such are the relationships that seem to exist in the Pythagorean tree (with respect to side length, area, and volume), and we will now prove them."


Structure
  • The Basic Description right now contains the description of how you guys made the Pythagorean tree. I suggested above that you could add to the Basic Description some content on the definition of what the 3D Pythagorean tree is. If you do that, then you could probably create a new section (probably under Basic Description) on how to build the 3D Pythagorean tree.
  • It is probably appropriate to move "The Edge Length Relationship" before "The Surface Area Relationship" section - it seems like you want to move up sequentially in dimensions, so start with edge lengths, then go to area, then go to volume.


Proof Strategy
  • There are a couple changes that I think need to be made to the proofs.
  • Your goal is to show the relationship between a larger square and the two smaller squares that follow it with respect to edge length, surface area, and volume.
  • I think it might facilitate the proofs if you could refer to a simple picture of just one iteration (with 3 squares) with the sides labeled a and c. This might not even need to be 3D for the sake of these proofs. Feel free to draw that up and upload it, or let me know if you would like me to do it.
  • One comment on proof strategy:
    • Say we want to prove a statement like A = B. In order to do this we need to take the expression A and show that it is equal to B. In other words, we need to take A (without an equals sign) and manipulate it so that it "looks like" B. If you start with the equation A = B (rather than the expression A), then you are already assuming what you want to conclude (that A is equal to B). Even college students frequently make this mistake of assuming their conclusion. It doesn't really mean that your math is wrong, but it is not quite in proper proof format.
    • I think the confusing part with these proofs is that the equations that you come up with are very similar to the Pythagorean Theorem, so the approach you guys have taken is to start with the Pythagorean Theorem and then manipulate it. Instead, you want to start with an expression for the edge length, surface area, or volume of either the big cube or the two small cubes, and substitute in the Pythagorean Theorem to show that it is equal to the edge length, surface area, or volume of what you are trying to show.
    • As an example, this is what you prove with volume:
2A^2 = C^2
2^{3/2}A^3 = C^3
2^{1/2}A = C
  • The math here is all correct, but it doesn't prove anything about volume; you've just manipulated the Pythagorean Theorem. And this is why you are getting the same relationship for edge length, surface area, and volume - because each time you start with the Pythagorean Theorem, which is an equation. I will come back to the volume proof later.
  • The other thing I want to comment on with respect to the proofs is the way that you prove your relationships.
    • What you are basically arguing is that
      • 1). the edge length of a cube is a factor of \sqrt{2} larger than the edge length of the succeeding cube,
      • 2). that the surface area of one cube is equal to the surface area of the two cubes that follow (in other words, there is a factor of 1), and
      • 3). that the volume of two equal-sized cubes is a factor of \sqrt{2} larger than the volume of the preceding cube (or, that the volume of the preceding cube is {1 \over \sqrt{2}} the size of the volume of the 2 equal-sized cubes).
    • These should be what you showed before with your chart at the beginning of the More Mathematical Explanation.
    • For instance, you could say that there is some constant k such that, for example, a^3 + b^3 = kc^3. In other words, the sum of the volumes of the latter two cubes is the volume of the former cube multiplied by some number. Then you would prove your relationship by showing that the constant k is \sqrt{2} or 1 or whatever.
    • For instance, you would prove the volume relationship this way:
Start with a^3 + b^3. Remember that we want to begin with an expression rather than an equation. We want to find k, so we want to manipulate this expression so that it only has a c^3 term multiplied by a constant. Then, by comparing the result to a^3 + b^3 = kc^3, we will know that the constant and k are equal.
Since, in our example, a = b, we have 2a^3 as the sum volume of the two latter cubes. This is equal to:
(2^{2/3}a^2)^{3/2} (Note that because this is an expression, it needs to remain equal to 2a^3. We cannot change the value of the expression. We have changed the exponents of the things in the parentheses by accordingly changing the exponent outside.)
Furthermore, this equals:
(2^{-1/3} \cdot 2a^2)^{3/2} (Here we have just worked inside the parentheses; the exponent on the outside has remained unchanged.)
Now, from the Pythagorean theorem with a = b, we have 2a^2 = c^2. Since 2a^2 is in the expression we have been working on, we can substitute this equality in:
(2^{-1/3} \cdot c^2)^{3/2}
Now we can get rid of the exponent on the outside of this expression:
2^{-1/2}c^3
So our constant k is equal to 2^{-1/2} = {1 \over \sqrt{2}}.
  • Note that in this example, we only changed one side of the starting equation (a^3 + b^3 = kc^3), and the starting equation was a relationship between the volume of the big cube and the two smaller cubes, rather than the Pythagorean Theorem. We used the Pythagorean Theorem to manipulate the expression we were working with, but we did not begin with it. Our conclusion is that the big cube's volume (c^3) is {1 \over \sqrt{2}} times the sum of the two smaller cubes' volumes.
  • In the rest of the proofs, we want to do the same thing. Start by setting up an equality like ka = c (for edge-length relationship) and show, using the Pythagorean theorem and by modifying one side of the equation, that k = \sqrt{2}. You will probably find it easiest to show this by modifying the side that does not contain k. Likewise for the surface area relationship, you would begin with k(6a^2 + 6b^2) = 6c. You would look at one side of this equation and then show that k = 1.


Future Content
  • Total volume or surface area after some number, say i, of iterations.
  • Total volume or surface area after infinite iterations
    • Does the tree have infinite volume? Infinite surface area? Or does it turn out to be finite?
  • If you guys are interested in these topics, you could probably find some online information. I will look into these at more depth to see how accessible they are.

Please let me know (by posting here or emailing me) if any of this does not make sense. I know the stuff about proofs is complicated and unfamiliar; working out formal proofs is hard to get used to.

--Gbrown2 15:12, 1 July 2013 (EDT)

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