Notebook Collection Part 2

In which I continue to share my collection of blank notebooks and journals.

These large notebooks have unconventional papers: Rhodia "DOTS" paper, Whitelines graph paper, Bob's Your Uncle "pretty vacant" dot paper. Writersblock dots, and a custom notebook from etsy.com

These are some of my large notebooks, the theme here is "unconventional paper" -- blank paper and lined paper are very nice, but when you spend as much time as I do staring at notebooks a change of scenery is nice.

This is what whitelines paper looks like.

Whitelines paper is a love or hate thing. I know many people who love it and just as many who can't stand it. I'm on the love side. I find it much easier to see my drawings and diagrams without black lines cutting through them. I have rejected many notebooks for having lines that are far too dark. To my eye dark lines look cheap, like those cartoon character notebooks you see in dollar stores.

The Rhodia Dot Pad was my favorite paper until I discovered Writersblok dot paper. As I mentioned in my previous post Writersblok is very much underrated and they make the best dot paper around. I'll show you why:

Three kinds of dot paper: Bob's Your Uncle, Rhodia, Writersblok

Here you see the inside of the three notebooks with dotted paper. The Bob's Your Uncle notebook is very fun, but the dots are so big that they don't really work for graphing. (Bob's Your Uncle graph paper notebooks on the other hand are amazing, very relaxed light green grid, I'd share but I filled them all up so quickly that I have no blank pages!) Rhodia is better, but the dots still stand out on the very white paper. But, Writersblok has it perfect! Look at how fine those dots are, they almost don't even show up in the photo!

What about the green "custom" number? It's a mix of different types of graph paper, dot papers and blank pages.

The green "Custom" notebook.

Next up? Hardcover books:

Hardcover blank books.

I keep my eye out for re-purposed journals with scientific or mathematical themes.

The cover is from an old book of mathematical tables, pages from the original are mixed in.

The burgundy recycled journal has a Coptic binding. (I'm not a huge fan of Coptic binding, but this one is very nice!)

"Mathematics" you'll find the same title on the spine.

Open it to find...

The "mathematics" book has pure white blank pages.

I don't know what to write in the "mathematics" book it is very formal and commanding isn't it?

Boorum Pease makes the most amazing record books.

This RECORD book has a lovely red satin bookmark. These can be found at old house sales, or on Amazon for around $20.

Record book open.

I don't often used lined pages but I'll make an exception. The numbered pages and "Table of Contents" page are the kind of little touches that make me fall in love with a notebook. The red and black record book to the right also has numbered pages....

I think I'm in HEAVEN!

... BUT instead it has quad paper! My favorite. I first discovered this notebook at a Manhattan stationary store. It was very dusty and I literally fished it out from behind a cabinet. It was torn in a few places, but the store owner still wanted $90 for it! I was so in love I almost bought it.

So, when I discovered I could get one on Amazon for $30 ... well that seemed like a deal. I want to take this one camping with me this summer.

I still have a lot more to share, but that's all for today! If you have notebooks that you love let me know in the comments.

Notebook Collection

There is nothing quite as exciting as a fresh blank notebook full of possibilities, well except, perhaps a full worn notebook filled with new mathematics! I can admit that I'm addicted to notebooks. Whenever I see one, with a new kind of color or material I buy it. As such, a shelf in our house is totally devoted to blank notebooks.

The Notebook Shelf

The Notebook Shelf

It was only a few months ago when I put them all in one place that it hit me how many I'd accumulated. These are all blank. I go through about one or two notebooks per week (of the Moleskine Cahier size) So, I have almost a whole bookcase full of full notebooks as well. I use Rhodia, Moleskine, Whitelines, Writer's block and other big brands quite often, In this post I'll focus on the more rare and unique stuff, the notebooks that make my hand tremble with excitement as I get ready to make that first mark on the unblemished page! The notebooks that I cherish, they are almost too good to write in, yet, I await the day that I mark them with such pent-up desire!

Okay, maybe I'm getting a little too excited here, on with the notebooks!

I like large notebooks best, they are best suited for lesson plans and proofs.

Although I like large notebooks best I do have a few small ones that I could not resist. Here we have: Quattero, a generic red notebook, an antique 1950s notebook, Writer's block "Dots" mini notebook and "Tidbit" Free Cut memo.

Quattero notebooks come in all sizes, they are a little pricy, but the paper is so smooth and thick it's worth it, the only thing I don't like about them is that the backs of the papers are blank, you only get the grid on one side, considering that the paper is bleed-proof this seems like a missed opportunity.

Here's what they look like inside: Writerblock "dots", Quattro, and the little red one.

Writersblok is one of the most underrated notebooks around. They are a less expensive version of the "Moleskine Cahier" but, even if they cost more than Moleskine I'd still take them over the Moleskine any day! Why? DOTS. Dots are the ideal method of ruling the blank page, grid paper is too distracting, lined paper looks awful if you stop to make drawings and graphs as often as I do. So dotted paper is one of the main things I look for in a notebook. The dots in the Writersblok notebooks are the best, they are whisper light, so that you'll almost forget they are there. Yet when you look at your pages you'll see that the lines are straighter, graphs are more accurate and the text is evenly spaced. I've been seeing more and more kinds of dot paper, I really hope it catches on!

Antique notebooks fascinate me. Especially those that have survived totally blank and untouched like this one for decades. I love that this little memo book has an even smaller notebook (not an address book!) tucked inside. If I ever think of the perfect use for it I will use it, but for now it remains untouched, it is one of my most cherished notebooks. (*sniff*)

Notebook within a notebook...

Notebook within a notebook...

Ever since I started loving blank books and shopping for them I have starting HATING address books, planners and photo albums. Why? Often I'll see a notebook on the shelf, the perfect size! The perfect color! I reach for it and.... it's a PHOTO ALBUM! ugh. So annoying. I end up wondering who are all of the people who like photo albums and address books so much, have they not heard of Flikr and gmail? I suppose they must be wondering what's with all of the blank books... who needs paper? I have an iPad.

(I will continue sharing my collection in the next post.)

Of yarn and math.

A collection of hand-spun yarn and my spindle. I did not spin all of this yarn, some is from etsy.com

Yarn has been on my mind. It's on my mind because I love to knit, but also because of this post I read some weeks ago on the mathematics stack exchange. The author presents a paper about an "optimal yarn ball" that looks like this:

via. http://arxiv.org/abs/1005.4609

Very cool, but it would be impossible to do that with real yarn! (Though, I doubt that was their goal here.) When wrapping yarn around a ball you cannot make sharp turns or the yarn will fall off. If we think of the yarn as a curve on the surface of the sphere, we would say it must have curvature less than some small constant m. (meaning the best fit tangental circle must have radius greater than \frac 1m)

The yarn should be wrapped evenly: it should not overlap itself too often. (This would lead to the yarn ball taking on a non-spherical shape over time. No good!) The distribution of the self-intersections should be of nearly consistent density over the surface of the sphere. The yarn will divide the surface of the sphere in to regions. If A_i, A_j are the areas of any two of these regions then for some small constant F, |A_i- A_j| < F, \forall i, j.

As the curve is extended further and further (wrapped around more times) the density of intersections should increase and A_i \rightarrow 0 for all of the regions.

I know one answer is a randomly generated curve that deviates from a straight path. Overtime it produces a perfect yarn ball. That's how I make my yarn balls, randomness produces even inward tension which forms a sphere.

But, I want to know if there is a non-random answer to this question.

First Thoughts:
In trying to solving this myself I thought it might be a good idea to project the sphere on to the plane, as a Riemann surface. But, since so many of the requirements focus on uniformity, this makes the problem a little strage. Observe: The area of the regions will need to *increase* as we move away from the unit circle toward infinity. If the curve ever passes through the north pole, then the plane version would shoot off to infinity. What I found even more weird is what would happen if the curve simply passed very near, but not through, the north pole. Then, on the plane, the curve would go very far from the origin and then loop back. By making the curve near enough to the north pole the loops can be as large as we please.

Since wrapping increases the density uniformly, if we observe the wrapping pattern projected on to the plane it would be like loopy knot that, over time, keeps casting bigger and bigger outlier loops, it would always grow and be un-bounded. (even without any intersection with the north pole!)
The requirement about curvature, would mean that the knot could be more curvy inside of the unit circle, and then grow less curvy as we move away from the origin.

Next to find such a curve... Maybe if I projected the following polar plot back on to the Riemann sphere?

The graph of the form r= \theta^2 \sin (k \theta) produced good-looking results for 0 < k <1... but all of the graphs are far too dense near the origin. Also, wrapping would start in a lop-sided manner, filling in the bottom of the sphere first then moving up... and never reaching the north pole! But, maybe there is some way to tweak this?

(This graph shows the kind of non-random pattern I have in mind for a solution. In a practical sense, a solution to this problem could be used to wind spherical balls of ropes, yarns, or cords in a factory setting. Now, I will think about how I would instruct a robot to wind a yarn ball... if the robot is unable to do things randomly. What would you tell the robot to do?)

This post grew out of a question I asked at the Stack Exchange.

Almost Convergence.

The title of my last post "Almost Cauchy Sequence" made me wonder if "Almost Cauchy" was already defined in some formal sense. I could not find much on "Almost Cauchy" ... but I did find out about "Almost Convergence." (I think I have ran in to this before... but it is very neat... observe:)

A sequence S = S_n is almost convergent to L if for any \epsilon > 0 we can find an integer n such that the average of n or more consecutive terms in the sequence is within \epsilon of L. Formally, \forall \epsilon > 0 \quad \exists N \ni \quad \displaystyle \left| \frac{1}{n} \sum_{i=0}^{n-1} S_{k-i} -L    \right| < \epsilon \quad \forall n > N, \; k \in \mathbb{N}. (I took this definition from here, and I wonder why they didn't mention that we need k \geq n or else we are looking at negative terms in the sequence... But, maybe there is an interpretation of something like S_{-3}? Or maybe the author thinks that would be obvious? I'm going to ask at the mathematics stack exchange!)

Almost Cauchy Sequence, unfinished business.

I just remembered a question from beginning analysis that my professor mentioned in class:

"Find a sequence A that is not Cauchy, yet any consecutive terms A_n, A_{n+1} will become arbitrarily close."

It was one of those questions that he mentioned in class for us to do on our own, but I never quite got to it. I was thinking about convergence and complete metric spaces today (and about unfinished business, as I've made a lot of my own since I started teaching.) And the question came back to me.

We seek a sequence, A, where \forall \; \epsilon > 0 \quad \exists N \in \mathbb{N} \;  \ni \; \forall n > N \quad |A_n - A_{n+1}| < \epsilon and \exists \epsilon > 0 \; \ni  \quad \forall  N \in \mathbb{N} \quad \exists m,n > N \quad |A_m - A_n| > \epsilon. (Writing the negation of what it means for a sequence to be Cauchy was a little more tricky than I thought...)

The natural log function is the most obvious answer. For large vales of n \ln (n) - \ln (n+1) is small. That is, \displaystyle \lim_{n \rightarrow \infty} \ln \frac{n}{n+1} = 0. This is true becuase after taking EXP of both sides we have: \displaystyle \lim_{n \rightarrow \infty}  \frac{n}{n+1} = 1

Next, it seems obvious that the natural log can't be Cauchy, but just to be certian if we let \epsilon = \frac{1}{2} then, \forall N \in \mathbb{N} if we choose some n > N and let m= ne (so, n,m > N) then

| \ln m - \ln n | = |\ln \frac{m}{n}| = |\ln \frac{ne}{n}| = 1 > \frac{1}{2} = \epsilon.

NOT Cauchy at all!

Ok, I imagine my professor was thinking of the natural log... but, what other functions also have this property? \sqrt[n]{x}? So, I guess this is really asking when \displaystyle \lim_{x \rightarrow \infty} f(x) - f(x+1) = 0 while \displaystyle \lim_{x \rightarrow \infty} f(x) = \pm \infty. It is necesairy but not sufficent for f'(x) > 0 \; \forall x and f''(x) < 0 \; \forall x , both will have the x-axis as a horizontal asymptote.

Nice to have finally gotten around to doing this problem! I think it will be good practice to find more unfinished business in my notes.

A good starting point.

Looking for a new area of mathematics to explore? If there is anything on this page that you have not studied, it might prove to be a good starting point.

It is a nice list.

Introducing the multi-zeta function!

Contemplating:

\frac{\pi ^{2n}}{(2n+1)!} = \displaystyle \sum_{j_1,...j_n=1 \atop j_i \neq j_k}^{\infty} (j_1j_2...j_n)^{-2}

My friend, and fellow math blogger, Owen, tried to tell me I was dealing with the multi-zeta function the other day, but the very general definition on WolframMathworld left me feeling a little mystified. I could see how the identities I was playing with fit in to it, but what was all of that other stuff (like \sigma_1,...,\sigma_k? I guess in my equation they are all 1? Why isn't it dealing with powers of the active variable in the product in the denominator? Why the extra level of subscripts?) So, I asked about it on stackexchange to gain more insight, and, thanks to Marni, I was promptly pointed to this lovely paper:

Now, I can see easily that the function is:

A(i_1, i_2, ... , i_k) = \sum_{n_1 > n_2 > \cdots > n_k > 1} \frac{1}{n_1^{i_1}n_2^{i_2} \cdots n_k^{i_k}}

With i_1, i_2, ... , i_k = 2.

More on this later.

Meaning one may repeat values,

Play with the follium!

I've discovered Wolfram Alpha's widget builder. It's very easy to make little searches that help focus your readers (or students) on certian topics in Wolfram's massive database. Use the above to investigate the follium ... as I mentioned in an earlier post.

Several series for powers of pi.

The Weierstrass factorization theorem gives us this identity:

\displaystyle \prod_{n=1}^{\infty} \left( 1 -\frac{q^2}{n^2} \right) = \frac{\sin(\pi q)}{\pi q}

Take the left side:

\left( 1 - \frac{1}{1}q^2 \right) \left( 1 - \frac{1}{4}q^2 \right) \left( 1 - \frac{1}{9}q^2 \right) \cdots

Thinking of this in a combinatorial way we can expand it. Each even power of q has a sum of combinations of the inverse squares as its coefficient. That is:

1 + \displaystyle \left(  q^2 \sum_{j_1=1}^{\infty} -j_1^{-2} \right) + \left( q^4 \sum_{j_1,j_2=1 \atop j_1 \neq j_2}^{\infty} (j_1j_2)^{-2} \right) + \left( q^6  \sum_{j_1,j_2,j_3=1 \atop j_i \neq j_k} (j_1j_2j_3)^{-2} \right) + \cdots + \displaystyle q^{2n} \sum_{j_1,...j_n=1 \atop j_i \neq j_k}^{\infty}( -1)^n (j_1j_2...j_n)^{-2} + \cdots

Now on the right side of
\displaystyle \prod_{n=1}^{\infty} \left( 1 -\frac{q^2}{n^2} \right) = \frac{\sin(\pi q)}{\pi q}

Expand using the Taylor series for the sine.

\frac{\sin(\pi q)}{\pi q} = 1 - \frac{(\pi q)^2}{3!} + \frac{(\pi q)^4}{5!} - \frac{(\pi q)^6}{7!} + \cdots

Equate the coefficients of the terms with the same exponent.:

  • 1  = 1
  • - \frac{\pi ^2}{3!} = \displaystyle \sum_{j_1=1}^{\infty} -j_1^{-2}
  • \frac{\pi ^4}{5!} = \displaystyle  \sum_{j_1,j_2=1 \atop j_1 \neq j_2}^{\infty} (j_1j_2)^{-2}
  • - \frac{\pi ^6}{7!} = \displaystyle \sum_{j_1,j_2,j_3=1 \atop j_i \neq j_k} - (j_1j_2j_3)^{-2}
  • \vdots
  • \frac{\pi ^{2n}}{(2n+1)!} = \displaystyle \sum_{j_1,...j_n=1 \atop j_i \neq j_k}^{\infty} (j_1j_2...j_n)^{-2}
  • \vdots

The second one is famous, more often written:

\frac{\pi ^2}{6} = 1 + \frac{1}{4} + \frac{1}{9} + \frac{1}{25} \cdots .
And the rest of these series I do not know so well. Do you know them?

Next step with the interesting difference from last night...

From last night's post:

\displaystyle  \prod_{j=1}^{\infty} (1 +x^j) - \displaystyle  \prod_{j=1}^{n} (1 +x^j)

This difference is the generating function for partitions into distinct parts from the set \{ n+1, n+2, ...  \}.

I think my next task is to graph the upper bound I found for these coefficients, then see how it differs from the actual number of partitions for this set. The difference should grow pretty quickly. I wonder how mathematica deals with generating functions. I know some of the popular functs are pre-loaded... But, can I give it an expression and have it list exponents for me?

So much to learn!