Author: thelabwithbradbarton

Fun with numbers and triangles

Fun with numbers and triangles

During my interview with Arthur Benjamin, he spoke briefly about the joy of discovering something in mathematics, even if one is relatively certain that the given relationship has been known for years, or even centuries. Along the same lines, I’ve been playing with right triangles of late.

They are called Pythagorean triples—three numbers, a, b, and c, where aSquared+bSquared=cSquared, and a b and c are all whole numbers. I wanted a way to generate them.

I started by noticing that the difference between two squares, such that one root is one less than the other, is always an odd number.

2*2=4
1*1=1
4-1=3
3*3=9
2*2=4
9-4=5
4*4=16
3*3=9
16-9=7

The result climbs up the odd number line, 3, 5, 7, 9, 11…

Another way to put this, is to say that the difference between consecutive whole number squares is on the line defined by 2x+1.

This can be shown algebraically:
(x+1)Squared-xSquared=2x+1
Expanding (x+1)Squared gives us (xSquared+2x+1)
So (xSquared+2x+1)-xSquared=2x+1
Add xSquared to each side and we get:
XSquared+2x+1=xSquared+2x+1
Take xSquared from each side and we get:
2x+1=2x+1
Take 2x from each side and we get
1=1
That’s about as true as numbers can get.

I’m not the first to have noticed that relation, but I did run across it independently.

Next, in order to confine the results to whole numbers, we substitute some whole number that is greater than or equal to 1 for x, which will give you side a.

2*2+1=3

Next we square a: 3*3=9
Subtract 1: 9-1=8
And divide by 2: 8/2=4
That gives us side b=4.
Next since we know the difference in length from b to c is 1, c=b+1: 4+1=5
3, 4, 5
3*3=9
4*4=16
9+16=25
5*5=25

Let’s try using 2 in place of x. 2x+1
2*2=4, 4+1=5, a=5
5*5=25, 25-1=24, 24/2=12, b=12
12+1=13 c=13
5, 12, 13
5*5=25
12*12=144
25+144=169
13*13=169

This is much easier to do in your head than it is to write it out.
3*2=6+1=7
7*7=49-1=48/2=24
24+1=25
7, 24, 25
4*2=8+1=9
9*9=81-1=80/2=40
40+1=41
9, 40, 41

When I looked it up, it turned out that Euclid used this trick to prove that the set of Pythagorean triples is infinite, since the set of odd numbers is infinite and you can use any odd number to generate sides a b and c.
Q.E.D.

Next, I got to wondering if you could generalize the method, so that you could generate triples when the difference between side b and c, c-b could equal any whole number.

Yes you can! If d=c-b than the line that your numbers need to sit on is (2*d)x+dSquared, where x is any whole number greater or equal to 0, unless d=1.

If d=1, the line is 2x+1. If x=0, you end up with 1, 0, 1. And while 1*1+0*0 does equal 1*1, that’s not really a triangle; it’s a line segment.

By the by, when you use d=1 you get 1*2=2 and 1*1=1, so the line is 2x+1, which we already knew.

Let’s try d=2.
2*2=4 and 2*2=4, so our line is given by 4x+4
Put 0 in place of x.
4*0=0+4=4, so a=4
4*4=16-4=12/4=3 so b=3
3+2=5 so c=5
4, 3, 5, which is our old friend 3, 4, 5

Using 4x+4 and replacing x with 1
4*1=4+4=8, a=8
8*8=64-4=60/4=15, so b=15
15+2=17
8, 15, 17

Using x=2, 2*4=8+4=12, a=12
12*12=144-4=140/4=35, b=35
35+2=37
12, 35, 37

Now let’s set d to 3
3*2=6 and 3*3=9, so the line is 6x+9
Starting with 0
6*0=0+9=9, a=9
9*9=81-9=72/6=12, b=12
12+3=15
9, 12, 15

X=1
6*1=6+9=15 a=15
15*15=225-9=216/6=36, b=36
36+3=39
15, 36, 39

X=2
6*2=12+9=21, a=21
21*21=441-9=432/6=72, b=72
72+3=75
21, 72, 75

In fact, there are simpler methods for making triples, but I found this all on my own, and I can’t even describe the joy of it, when the numbers click together and it all makes sense.

As any whole number greater than 0 can be substituted for d, and the set of whole numbers greater than 0 is infinite, this adds another layer of infinity on top of the one Euclid found, just in case one infinity wasn’t enough
Q.E.D.

Ep 27: It doesn’t have to be true to be useful

Ep 27: It doesn’t have to be true to be useful

It doesn’t have to be true to be useful

Today we talk about the placebo effect—when an inactive substance can mimic the action of medicine or other therapies.

Here’s a review of many studies and experiments done on the placebo effect.

How Placebos Change the Patient’s Brain

And here’s a link to episode 25, which I reference toward the end of today’s episode.

Ep 25: It isn’t all about positive thinking

ep 26: Lost in the mountains, with no magic

ep 26: Lost in the mountains, with no magic

Lost in the mountains, with no magic

There doesn’t seem to be any one thing that could transform you into a genius, and the search is complicated by exaggeration and fabrication within the research.

Here’s a paper on where your idea of your self, and where the circuits that let you regulate said self live.

Neuroscience of Self and Self-Regulation

Here’s an article on neural plasticity, critical periods of development, and increasing the ability of the adult brain to learn.

Re-opening Windows: Manipulating Critical Periods for Brain Development

Some days are better than others

Some days are better than others

I just finished this morning’s session with the vOICe. My performance was less effective than the last few sessions, so far as reach and grab, or reach and get close. I only had visual qualia a few times. More than once, I “saw” it, and then reached for the wrong spot, only to find that it was where I’d “seen” it. But the visual qualia came and went so quickly that it was difficult to localize the target. What’s more, when I started, I kept seeing several targets on different spots of the bed, ghost images.

I could avoid posting about sessions that don’t go well, but somewhere out there, someone else who is training with the vOICe, or considering using it needs to know that some days will go better than others. What’s more, gaining facility could take months, or years.

If you consider neural network entrainment, one could assume that today was about testing the model that my brain has built up so far.

I’m using the vOICe and my own experience as part of the overall research efforts, attempting to find ways to improve my ability to use my own mind and brain in a more optimal fashion. However, because I’m trying a number of different things, if I do suddenly find myself learning at a considerably faster rate than the average, it will be difficult to tell which thing or combination of things caused that, or if I just happen to luck out and be unusually good at learning this task, just because of the way I’m wired, regardless of my approach. I could also end up being slower than most, but still be better than I would have been without the experiments. Or perhaps the experiments might be counter-productive. There are so many confounding factors.

Meanwhile, I find that my instinct is to inhibit my emotional reaction, to try and hold down my level of frustration. This is something I do more or less automatically. Yet those with savant syndrome often have trouble regulating their emotions. That’s also true when one considers the tortured artist, mad scientist, or even children. Perhaps, the neural chemical reaction associated with frustration helps to refine the neural patterns that are being build up as my brain models the process. Perhaps I would learn better if I did get upset, so long as the frustration doesn’t interfere with my motivation and push me away from continuing to practice. All respect to Shawn Achor and the positive psychology folk with their “happiness advantage,” but just because something feels unpleasant, doesn’t mean that what’s happening is “bad.”

Oh, for a budget, and willing participants.

Meanwhile, more Tetris effect. In fact, as soon as I was done and sat down to turn on my computer and write this, I started to hear the sounds of the vOICe, so at least some automatic processing is still going on. Here’s one place where some insight strikes me as useful. I could fight it—try and push the back of the head noises completely out of my head. Instead, on the theory that more processing is better, I do my best to let it run.

An auditory Tetris effect

An auditory Tetris effect

It’s called, the Tetris effect.

People who have played the game Tetris often find that they will start to see those falling blocks as they doze off, as they are just waking up, or even when they blink from time to time as they go about their lives. Though named after one rather addictive video game, the effect isn’t limited to that game. In fact, just about any activity can leave echoes playing in your mind. If you spend the day playing on the beach, you can find yourself feeling as though your body is being pushed and pulled by waves as you fall asleep that night, an echo off the sun and fun you had while playing in the shallows and serf. In similar fashion, a day on a small boat can leave the solid ground seemingly shifting and rolling under your feet, or if you play drums all damn day, just about every sound you hear starts to seem like it’s part of a persistent and complex rhythm.

Here’s a two-and-a-half minute YouTube video that both covered the subject, and made me smile.

The Tetris Effect

This morning, I was working with the vOICe, yet again. As I mentioned in the first post on the subject, I was getting some of those visual qualia again. This time, sometimes when I saw the target, where I saw it was rather far away from where the little white cube actually was. Both the occurrence of visual qualia and the accuracy of said qualia correlated with how well I was able to trust the back of my head. That is to say, the harder I tried, the worse it went.

After spending roughly 30 minutes with it, I moved on with my life. Later, I was on my computer, looking up something entirely different. That’s when I noticed that in the back of my head, I was hearing the sounds of the vOICe, including the beep like sound that corresponds to the presence of the cube. In fact, once, as I leaned back in my chair, puffing my air out through my lips, letting my mind rest for a moment after reading whatever the hell I’d just been reading, I noticed that I both heard it, and sort of saw it, in the same rather vague shadow like way that I see the target when I get visual qualia. For a moment, the “sight” of the cube moved up and down and side to side, and the sound of the cube did the same within the sound scape. It’s much like having a song stuck in your head, or seeing scenes from a movie you’ve watched recently behind your eyes.

I suspect this is my brain, replaying the sensory memory in order to train my neurons to be able to model the real-world phenomenon. A quick Google search just before I wrote this and I find that most researchers seem to think the same thing.

Go brain go!

Who turned up the world?

Who turned up the world?

I was chatting on Twitter back and forth with @seeingwithsound about the vOICe. My Twitter handle is @lobbyandlab if you’d like to join the conversation. I mentioned that my echo location seemed to have improved since I began working with the vOICe. @seeingwithsound replied that though it was only anecdotal evidence, he’d noticed that his sensitivity to subtle sounds seems to have increased.

Wherein it is still anecdotal evidence for you and the rest of the internet, it is direct evidence for me.

Quiet things seem to be subjectively louder. I’m not certain that’s an accurate way of phrasing it–what I do know is that I’ve been mildly surprised by a soft sound from time to time. It happened this morning when I rolled over in bed, and heard something that made me stop and go, what was that? It turned out just to be the sound of one blanket rubbing against the other, a sound that normally doesn’t intrude its way into my awareness. Another time I was unconsciously tapping my toe, and again, my focus was involuntarily drawn to the surprisingly loud seeming sound of my sock-clad foot moving around in my shoe.

Presumably, there has been an increase in the firing rate of some of my auditory nerves, or those that receive auditory nerve signals. I may also have formed more connections, so that more neurons are firing in concert for a given mild sonic stimulus.

I wonder if anyone has figured out how the brain judges the relative volume of sounds. I’ll have to look it up.

Ep 24: Which part of the brain, off on and when

Ep 24: Which part of the brain, off on and when

Which part of the brain, off on and when

There is nothing simple about the brain. Wherein certain skills can be improved by shutting down one part, those same skills may require increasing activity in another.

Here’s a collection of short videos on the flow state of consciousness.

The Optimized Brain: A Workshop on FlowStates

Here’s a paper on the results of scanning brains of jazz musicians during improvisational playing.

Neural Substrates of Spontaneous Musical Performance: An fMRI Study of Jazz Improvisation

And here’s a ted talk by the author of the above paper on the same subject.

Your brain on improv

I’m not sure how I feel about it when scientist rap. Not because there’s anything wrong with rap/ just that they’re generally so very very bad at it.

Ep 23: As I was saying

Ep 23: As I was saying

As I was saying

While it has been fun to talk about the vOICe and sensory substitution, it’s time to return to the brain, how it works, and how one might discover how best to use one of your very own.

Here’s a blog post, with no podcast episode, about the vOICe, and my attempt to apply some of what I’ve learned and/or guessed about the brain.

A modified reaching exercise

And here’s the panel on genius by the world science festival.

Beautiful Minds: The Enigma of Genius

A modified reaching exercise

A modified reaching exercise

Over the weekend, I did a good deal of research. I was trying to track down which parts of the brain may interfere with certain skills. But I wanted more than just which bit does what; I wanted to know the subjective feel. How does it feel when that part of the brain is active, and can I consciously inhibit that part of the brain in order to improve performance?

I’ve been working with something called the “vOICe.” For background, you can checkout episode 19. Based on what I’ve been learning, I decided to slightly modify an exercise from the vOICe training manual—reach and grab.

If you consider mnemonic techniques, the approaches involve attaching things that are difficult for your mind to recall, to things that are easy. you tie the new information to structures within your brain/mind that already are in place. It occurred to me that I might be able to do that with the soundscapes. Perhaps, I could use methods that are already in place within my brain to help me keep track of where objects are in relation to my body while absorbing the new source of information, so that the new form of processing could piggyback on methods that are already in place.

Thus, I would set the target I was reaching for, “A small cube of white modeling clay,” down on the surface of my bed. Then I would use the vOICe to listen to where it was, when I already knew where it was. I did this multiple times, placing the target in different places, listening with the camera at different angles, and reaching out to touch the target with either hand.

When I’d toss the cube so that it would bounce slightly and land somewhere I wasn’t aware of, I’d find it in the soundscape, and then reach out for it. If I missed, even by just a little, instead of picking it up, I’d step back, look at it with the vOICe again, and reach out to touch it several times. It’s much the same as setting down the target somewhere on purpose, only it starts without knowing exactly where the target is, until you’ve already touched it.

Going by the notion that my forebrain might interfere with the sensory processing that usually takes place mostly in the back brain, I changed the way I listen to the soundscapes. I don’t try and hear where it is. I don’t focus too strongly on the soundscape at all. Instead, I’m simply aware of the soundscape, and I avoid trying to understand how and why it begins to work.

I was able to have some visual qualia. Visual qualia is the way that things look like they look; the steady seeming sensation of actually seeing something, and it turns out to be independent of the way the information reaches your mind.

I’d be listening to the voice, and after a time, like seeing a little spark or glint, I’d “see” the little cube appear in my view. Interestingly, I often missed by a tiny bit, even when I’d “seen” the target, by about the same amount I’d tend to miss without the visual qualia. When that happened, sometimes the little image would jump slightly as I noticed where it really was, but more often, it would simply vanish.

There were other times when the subjective “sight” of the target was completely correct, and I could just reach out and grab it.

Unfortunately, I haven’t been keeping track of my error rate, and I have nobody around to hold the clipboard, as it were.

I’m putting terms like “look,” and “see,” in quotes, because my eyes no longer work well enough to provide the “sight” of the target. However, as I had enough vision in the past to know what looking at something looks like, I can say that it’s the same. I see it, like seeing it looks.

As I’ve been totally blind for around 6 or 7 years now, I’m likely to be using the visual cortex to help me model the world around me without vision. Looking with the vOICe when I already know where it is should help integrate the newer source of visual information into that process.

I imagine the mental balancing act of listening without straining is something that most users of the vOICe figure out instinctively, but I thought I might be able to accelerate the process by doing it deliberately. I believe it to be a mistake to try and imagine what it might look like; that strikes me as building a model with the wrong part of the brain, and I suspect doing so interferes with building the model in the parts of your brain that do this sort of thing automatically.

Sure, let’s separate mental processing into deliberate and automatic activity, based on how conscious you are of the process. I submit that deliberate mental activity can block or interfere with automatic mental activity.

I’m slowly putting together some idea on how to better allocate mental resources for a given task, and this is merely the first experiment. I have no control, and no way to avoid observer biases, and this is only the very first try. Stay tuned.