Some musings on materials science

I’m working on writing up a longer post on a energy conservation experiment we did earlier in the year, but along the way, I’ve come up with a question that’s left me somewhat puzzled. How do we explain the fact that a steel spring flexes and returns to its original position when pressed, while a glass plate, when compressed often shatters, and a piece of clay deforms.

This seems like a very basic question in Materials Science, but for me, the answer has always been “atomic structure”—molecules in steel are very tightly arranged in a crystalline structure, while those in glass is much more disordered, and clay is even more disordered. It wasn’t until some researching today that I even realized that modeling clay is made up of a mixture of minerals (mostly aluminum silicon oxides), oils and waxes.

I get that the large scale crystalline structure of steel can explain its flexibility—the bonds between atoms each act like tiny springs, and thanks to the regular arrangement of atoms in the material, it is easy to distribute a load on the surface by compressing each bond” by only a tiny amount (springs in series add).

In glass, it’s much more difficult, since there’s no large scale structure. Following a line below a load doesn’t follow a regular arrangement of atoms and bonds, instead there can be places where the bonding changes quite rapidly between regions, and thus the same support force might require a much larger deformation than a neighboring region. This deformation can sometimes be large enough to actually break bonds between molecules, and can lead to a cascading failure as suddenly there are now fewer neighbors bonded to share the stress, causing the stress to increase on each neighbor, possibly past the breaking point.

Clay seems very similar to glass, in that an amorphous arrangement of regions of minerals oils and waxes. The major difference between clay and glass seems to be that when one of these failures occurs that would lead to cracking, the faille can often be contained since the the oils and waxes are fairly fluid, and can allow regions under stress to slip past one another, deform, and redistribute the load in a way that doesn’t lead to a cascading failure or shattering like glass.

Once the load is removed in these three cases, the steel returns back to its original shape, so all of the energy stored in compressed bonds is released to other forms, thus explaining its highly elastic nature. In deforming the clay, some bonds end up permanently deformed due to the shifting of regions, so clay is very inelastic.

One other interesting thing I discovered is that it is possible to make glass more impact resistant, and the way this is accomplished is by replacing the smaller sodium ions in the glass with larger potassium ions. This stresses the top layers in the glass in such a way that they are better able to withstand compressive stress. I’m not quite sure I follow this—is part of what is being accomplished here is that we are making sure that the top layers have more organized large scale structure? I think this is true, but there must also be more to it. Or are the larger potassium atoms better able to act as springs—bonds between them must be stretched further in order to break?

I’d love any feedback to help me flesh out misunderstandings here.

After a nudge from Justin, I decided to go back and look at the relationship

$$N=\frac{n^2-n}{2}+nx$$

What is it about some numbers (very nice numbers) that allow for multiple solutions of \(n\) and \(x\)?

For example, 45 is very, very, very nice, with 4 different \(n,x\) solution pairs, while 43 is only nice, with one solution pair.

If you plot the curves in Desmos, you can see how similar the two curves are, yet the curve for 45 has 4 integer pairs on the curve in the 1st quadrant, while the curve for 43 has only 1. 

I spent a lot of time trying to solve this equation for n, and after a lot of frustration, I tried the much simpler approach of solving for x, which yields an expression for the starting term of an integer sequence, given the nice number \(N\), and the number of terms, \(n\):

$$x=\frac{N}{n}-\frac{n-1}{n}$$

Since x must be an integer, we can place some restrictions on possible values for \(N\) and \(n\). Here are three separate possibilities:

\(n=2\) and \(N\) is odd. In this case \(\frac{N}{n}\) will have a remainder of \(\frac{1}{2}\) and \(\frac{n-1}{n}=\frac{1}{2}\), making \(x\) an integer. This tells us also that all odd numbers are at least nice, and can be written as the sum of two consecutive numbers. 
\(N\) is odd and \(n\) is an odd factor of \(N\). In this case, \(\frac{N}{n}\) will be an integer and \(\frac{n-1}{n}\) will be an integer. This tells us that odd numbers with a large number of factors are nicer. This also helps to explain why a prime number, like 43 can only be written one way as the sum of two consecutive integers.
\(n\) is a multiple of 2 such that \(\frac{N}{n}\) has remainder \(\frac{1}{2}\). In this case, \(\frac{n-1}{n}\) will also have remainder \(\frac{1}{2}\), and the result for \(x\) will be an integer.  

		
This seems like very good progress—I think I can now predict how nice \(N\) will be on the basis of its factorization. Still, I think there must be a way to simplify this even further.

After a nudge from Justin, I decided to go back and look at the relationship $$N=\frac{n^2-n}{2}+nx$$ What is it about some numbers (very nice numbers) that allow for multiple solutions of \(n\) and \(x\)? For example, 45 is very, very, very nice, with 4 different \(n,x\) solution pairs, while 43 is only nice, with one solution pair. If you plot the curves in Desmos, you can see how similar the two curves are, yet the curve for 45 has 4 integer pairs on the curve in the 1st quadrant, while the curve for 43 has only 1. I spent a lot of time trying to solve this equation for n, and after a lot of frustration, I tried the much simpler approach of solving for x, which yields an expression for the starting term of an integer sequence, given the nice number \(N\), and the number of terms, \(n\): $$x=\frac{N}{n}-\frac{n-1}{n}$$ Since x must be an integer, we can place some restrictions on possible values for \(N\) and \(n\). Here are three separate possibilities:

  1. \(n=2\) and \(N\) is odd. In this case \(\frac{N}{n}\) will have a remainder of \(\frac{1}{2}\) and \(\frac{n-1}{n}=\frac{1}{2}\), making \(x\) an integer. This tells us also that all odd numbers are at least nice, and can be written as the sum of two consecutive numbers.
  2. \(N\) is odd and \(n\) is an odd factor of \(N\). In this case, \(\frac{N}{n}\) will be an integer and \(\frac{n-1}{n}\) will be an integer. This tells us that odd numbers with a large number of factors are nicer. This also helps to explain why a prime number, like 43 can only be written one way as the sum of two consecutive integers.
  3. \(n\) is a multiple of 2 such that \(\frac{N}{n}\) has remainder \(\frac{1}{2}\). In this case, \(\frac{n-1}{n}\) will also have remainder \(\frac{1}{2}\), and the result for \(x\) will be an integer.
This seems like very good progress—I think I can now predict how nice \(N\) will be on the basis of its factorization. Still, I think there must be a way to simplify this even further.

Musing on very nice numbers

Is it possible to test whether a number is very nice? Is 45 very nice?

In order for a number to be very nice, we must be able to write it as two different sequences of integers, each of different length. Let’s say one sequence starts at \(x\) and is \(n\) terms long, while a second sequence starts at \(y\) and is \(m\) terms long.

Then it must be that:

$$\begin{aligned}
\frac{n^2-n}{2}+nx&=\frac{m^2-m}{2}+my\\
n^2-n+2nx&=m^2-m+2my\\
n^2+n(2x-1)&=m^2+m(2y-1)\\
n(n+2x-1)&=m(m+2y-1)
\end{aligned}
$$

I realized that this must also mean that:

$$\frac{n}{m}=\frac{m+2y-1}{n+2x-1}$$

And this indeed is true for the very nice numbers that I’ve found and tested, but I don’t see how this gets me any closer to testing if 1234 is very nice, since I have no way in advance of knowing values for \(n,x,m,\;\textrm{or} \;y\).

Proof of why powers of 2 aren’t nice

In my last post exploring nice numbers, I seemed to stumble upon the idea that powers of 2 are not nice numbers, and perhaps, the only not nice number.

In this post, I’d like to prove that no power of 2 can be written as a nice number. Much of my thinking here comes from a great conversation with Justin.

Let’s start with our expression for nice numbers. All nice numbers \(N\) can be written with \(n\) terms starting at \(x\).

$$N=nx+\frac{n(n-1)}{2}$$

Let’s try to prove that \(N \neq 2^y\) where \(y\) is an integer. Let’s simplify a bit and consider only nice numbers starting where \(x=1\).

$$2^y=n+\frac{n(n-1)}{2}$$

If we expand the right hand side this, we get

$$\begin{aligned}
2^y&=n+\frac{n^2}{2}-\frac{n}{2}\\
&=\frac{n^2}{2}+\frac{n}{2}\\
&=\frac{n(n+1)}{2}\\
\end{aligned}$$

Now let’s take the logarithm of both sides:

\[\begin{aligned}
\log 2^y&=\log\left(\frac{n(n+1)}{2}\right)\\
y\log 2&=\log n + \log(n+1)- \log 2\\
\end{aligned}\]

Solving for y:

$$\begin{aligned}
y&=\frac{\log n+\log(n+1)}{\log 2}-\frac{\log2}{\log2}\\
y&=\frac{\log n+\log(n+1)}{\log 2}-1\\
\end{aligned}$$

y must be an integer, and this means that we require \(\frac{\log n+\log(n+1)}{\log 2}\) to be an integer. However, it’s not possible for \(\log n\), \(\log (n+1)\), and \(\log 2\) to be all be rational, much less integers, so we proven that no nice number starting with 1 can be written as an integer power of 2.

Justin nudged me toward a much more elegant way of solving this, starting with the factored

$$
\begin{align}
2^y&=\frac{n(n+1)}{2}\\
2^{y+1}&=n(n+1)\
\end{align}$$

This can’t be true, since \(n(n+1)\) must be the product of an even number and an odd number, which can never equal a power of 2. This is much more beautiful, and it reminds me that my approach to math is still often rooted in clumsily applying the tools and algorithms I learned in high school: “when you see a variable in the exponent, take a logarithm,” and “factor to make the logarithm rules easier to apply.”

I think I can extend this to show a nice number with any starting term can’t be written as a power of 2.

$$\begin{align} 2^y&=\frac{n^2}{2}+\frac{n}{2}+nx\\ 2^{y+1}&=n^2+n+2nx\\ &=n^2+n(2x-1)\\ 2^{(y+1)}&=n\left(n+\left(2x-1\right)\right) \end{align}$$

We can examine the for cases for even and odd \(n\) and \(x\). In all cases \(2x-1\) will be odd. When added to \(n\), this will be even when \(n\) is odd, and odd when \(n\) is even. When we multiply by \(n\), we will have a product of an even and an odd number in all cases, which can’t be a power of 2.

ps: I apologize for the weird formatting of the multiline equations. For some reason, MathJax and Tumblr don’t seem to be supporting multiline equations at the moment. Note: I fixed it by adding 3 slashes at the end of the line, since tumblr interprets two slashes as a linebreak. 

Let’s call a number n nice if it can be expressed as the sum of two or more consecutive positive integers. For example, the expressions \( 5=2+3\) and \(6=1+2+3\) so that 5 and 6 are nice numbers. Which numbers are nice? Justify your answer. Some numbers are “very nice”, in a sense that they are nice in more than one way. For example, 15 is very nice because \(15=1+2+3+4+5=4+5+6=7+8\). Which numbers are very nice? Explain. For a nice number,\(N\) that is the sum of two consecutive numbers, we can see $$N=x+(x+1)=2x+1$$ For 3 terms $$N=x+(x+1)+(x+2)=3x+3$$ if we generalize this pattern we can say a nice number with \(n\) terms can can be written $$N=x+(x+1)+(x+2)\cdots+(x+n-1)=nx+\sum_{i=1}^{n-1}=nx+\frac{n(n-1)}{2}$$ Where the second term is itself a nice number, so long as \(n>2\). This general expression simplifies to the two equiations above for \(n=2\) and \(n=3\). To determine what nice numbers exist that are the sum of 2 integers, we can simply graph the function \(y=2x+1\) and look for all of the integer values of x that will produce integer values of y. \(2x+1\) is also an expression for odd integers, so all odd integers must be nice. For nice numbers that are the sum of 3 integers, we have \(N=3x+3\) and this will produce multiples of 3 starting at 6: (6, 9 ,12, 15,…). It seems like this creates a pattern that nearly covers all of the integers as a nice number of some type. My next step was to write a program to find all of the nice numbers, and here is the program as a gist. When I compute all the nice numbers between 1 and 100 that can be written as the sum of 2, 3, 4 or 5 integers, I get the following list:

3, 5, 6, 7, 9, 10, 11, 12, 13, 14, 15, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 29, 30, 31, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 45, 46, 47, 48, 49, 50, 51, 53, 54, 55, 57, 58, 59, 60, 61, 62, 63, 65, 66, 67, 69, 70, 71, 72, 73, 74, 75, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 89, 90, 91, 93, 94, 95, 96, 97, 98, 99, 100,

What seems most interesting about this list is how few numbers there are that aren’t nice. They are:

2, 4, 8, 16, 28, 32, 44, 52, 56, 64, 68, 76, 88, 92,

Aside from the obvious evenness of these, we see that all powers of 2 aren’t nice, and this continues to for powers of 2 up to 1024. I think it might be good to come up with an algebraic proof of this, as well look for a pattern that might explain why some of the other numbers that aren’t powers of 2 aren’t nice. On other thing I found in with my program was that if we classify numbers in the following way: nice: one consecutive run of integers sum to the number very nice: two consecutive runs of integers sum to the numb very very nice: three consecutive runs of integers sum to the number very very very nice: four consecutive runs of integers sum to the number There are no very very very nice numbers that less than 500005. It seems like this could also be proven algebraically.

Let’s call a number n nice if it can be expressed as the sum of two or more consecutive positive integers. For example, the expressions \( 5=2+3\) and \(6=1+2+3\) so that 5 and 6 are nice numbers. Which numbers are nice? Justify your answer. Some numbers are “very nice”, in a sense that they are nice in more than one way. For example, 15 is very nice because \(15=1+2+3+4+5=4+5+6=7+8\). Which numbers are very nice? Explain. For a nice number,\(N\) that is the sum of two consecutive numbers, we can see $$N=x+(x+1)=2x+1$$ For 3 terms $$N=x+(x+1)+(x+2)=3x+3$$ if we generalize this pattern we can say a nice number with \(n\) terms can can be written $$N=x+(x+1)+(x+2)\cdots+(x+n-1)=nx+\sum_{i=1}^{n-1}=nx+\frac{n(n-1)}{2}$$ Where the second term is itself a nice number, so long as \(n>2\). This general expression simplifies to the two equiations above for \(n=2\) and \(n=3\). To determine what nice numbers exist that are the sum of 2 integers, we can simply graph the function \(y=2x+1\) and look for all of the integer values of x that will produce integer values of y. \(2x+1\) is also an expression for odd integers, so all odd integers must be nice. For nice numbers that are the sum of 3 integers, we have \(N=3x+3\) and this will produce multiples of 3 starting at 6: (6, 9 ,12, 15,…). It seems like this creates a pattern that nearly covers all of the integers as a nice number of some type. My next step was to write a program to find all of the nice numbers, and here is the program as a gist. When I compute all the nice numbers between 1 and 100 that can be written as the sum of 2, 3, 4 or 5 integers, I get the following list:

3, 5, 6, 7, 9, 10, 11, 12, 13, 14, 15, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 29, 30, 31, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 45, 46, 47, 48, 49, 50, 51, 53, 54, 55, 57, 58, 59, 60, 61, 62, 63, 65, 66, 67, 69, 70, 71, 72, 73, 74, 75, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 89, 90, 91, 93, 94, 95, 96, 97, 98, 99, 100,

What seems most interesting about this list is how few numbers there are that aren’t nice. They are:

2, 4, 8, 16, 28, 32, 44, 52, 56, 64, 68, 76, 88, 92,

Aside from the obvious evenness of these, we see that all powers of 2 aren’t nice, and this continues to for powers of 2 up to 1024. I think it might be good to come up with an algebraic proof of this, as well look for a pattern that might explain why some of the other numbers that aren’t powers of 2 aren’t nice. On other thing I found in with my program was that if we classify numbers in the following way: nice: one consecutive run of integers sum to the number very nice: two consecutive runs of integers sum to the numb very very nice: three consecutive runs of integers sum to the number very very very nice: four consecutive runs of integers sum to the number There are no very very very nice numbers that less than 500005. It seems like this could also be proven algebraically.

Bain: some audacious goals

Reading page 33 of What the Best College Students Do. Talking about Jeff Hawkins, pioneer in computer industry:

By the time Jeff entered Cornell as an 18 year old freshman, he had made a list of four great questions he wanted to pursue. First, why does anything exist? “Nothing seems more plausible than something,” he explained long after he had fathered the first successful mobile computing device and helped build Palm and Handspring into billion dollar corporations. Second, given that a universe does exist, why do we have the particular laws of physics that we do? Why is it that we have an electromagnetic field, or that $$E=mc^2$$? he mused. Third, why do we have life, and what is its nature? Finally, given that life exists, what’s the nature of intelligence? … Jeff received good grades, but never placed at the top of the class. “I did what I have to do in a class,” he said, “but I didn’t freak out about making the best grades.” He usually sat in the front row, paid attention, and did the work, but he focused on what fascinated him. Because simple answers never satisfied, he probed for deeper explanations. “In magic, that meant asking not just how the trick was done, but how anyone could be fooled by it.

Pretty astounding questions and practices for an 18 year-old.

Bain: key difference between mediocre and highly successful students

Another great quote from Ken Bain’s What the Best College Students Do (page 20):

One of the major differences we found between highly successful students and mediocre ones: average students think they can tell right away if they are going to be good at something. If they don’t get it immediately, they throw up their hands and say “I can’t do it.” Their more accomplished classmates have a completely different attitude—and it is largely a matter of attitude rather than ability. They stick with assignments much longer and are always reluctant to give it up. “I haven’t learned it yet”,” they might say, while others might cry, “I’m not good at” history, music, math, writing or whatever.

Reading What the Best Students do by Ken Bain

Here’s a great quote from What the Best College Students Do, by Ken Bain, p 4. Discussing what growth means to different students…

To only a few, Barker concluded, “growth is the discovery of the dynamic power of the mind.” It is discovering yourself, and who you are, and how you can use yourself. That’s all you have. Baker emphasized that in all of human history, no one has ever had your set of body chemistries and life experiences. No one has ever had a brain exactly like yours. You are one of a kind. You can look at problems from an angle no one else can see. But you must find out who you are and how you work if you expect to unleash the power of your own mind.

What’s the difference between identifying and non-identifying relationship

I had this question today, and stack overflow had a great answer.

simple way to create random string of letters in python

Quick tip on how to make a random stream of characters in python:


import random
import string

randomstring = ”.join(random.choice(string.letters) for x in xrange(5))