How Many Of You Had To Teach Yourself The Math?

It may be optimum for the serving size / value for most fish. Once a size has been committed to change is expensive. I've seen the old pre freezing machinery in fish plants in Alaska and Scotland and they weren't going to deal with different sizes.

The tuna can has been deal its death blow by the vacuum packed foil pouch. Now IF that form didn't cost more...

That may be true. But it doesn't really matter. What counts is having fun with the math, yes?

:)

Jon

How would you drink out of pop top tuna can? Its too wide to get a good grip when it sweats, and too shallow to keep from pouring it on yourself. With Carpal Tunnel Syndrome, I have enough trouble holding a regular pop can, so much that I have do drink from a mug with a handle.

How it ought to be done:

An expert in ergonomics studies multiple shapes as various humans drink from them and selects an optimum design.

How it is done:

Someone looks at it and says, "That looks about right".

Some things are so simple that it doesn't need a committee. Anyway, the size and shape of pop cans was selected long before the concept of ergonomics. They chose what they could make. Do you remember the real early steel soda cans? They didn't last very long, before the combo can with an aluminum body and a steel top came out, then finally the all aluminum can.

Soda cans are shaped for drinking from.

Bye. Jasen

Absolutely! Well, unless you're in the can business I suppose.

Tim

-snip-

Thanks Tom! I'll have to check it out when I get home.

I'm still plugging away at the college text books i have, but it never hurts to double one's efforts.;)

I find it intriguing that math can become its own 'alternate reality', but i seriously doubt that I'll take it that far. You never know, though, heh.

Thanks again!

-phaeton

It's a different universe than what we experience, directly. You experience it entirely within your mind, though there are real effects at a macro scale in our universe that are 'mathematical' and directly experienced (kind of) -- I'm referring to our sensations of light, which is highly mathematical in behavior and where some of that math behavior can be experienced rather directly, without having to delve into micro-scale atomic effects indirectly observed primarily through theories in science.

Think of it kind of as a landscape and world of its own. Highly unified and coherent, with rules that make sense and are consistently applied. It's knowable. But there are vast reaches of unexplored areas, too. As well as intimate details within more known areas still as yet undiscovered.

You can get lost in this universe. Many do.

A problem posed, at least as early as 450 BCE or so, was expressed by the Greek, Protagoras. He felt that there were many hindrances to knowledge of gods, for example -- the obscurity of the entire subject area born of the fact that we simply aren't gods ourselves as well as our brief lifetimes. He concluded that there could never be any absolute truths or eternal standards of right and wrong and that in the end, the only thing we could say would be particular truths that would be valid only for the individual attempting to know them. But nothing universal. He was one of the Greek Sophists of the time.

A few centuries later, the Skeptics had added their own twist to this

-- that all knowledge is derived from sense perceptions and therefore _must_ be limited and entirely relative. They then deduced from this realization that people cannot ever prove anything at all. Since it is given that our sense perceptions are organic, inherently flawed, and may deceive us, no truths are certain. They then argued that the rational course of life is to always suspend judgement. They felt that we could attain peace of mind only by abandoning a fruitless search for what cannot ever possibly be had.

With this in mind, and thinking closely about our sense perceptions as organic, imperfect, and quite likely giving us false information... then what is the solution?? How can anything at all be known?

But let me even pose this as a still more difficult puzzle. When we try and generalize about things, say to elicit the "true nature" of related things let's say, how do we do that? We use our own experiences to guide us. In other words, we might describe one thing in terms of our sensory impressions of something else. "The Earth is like a ball," for example, taking something very difficult to fully apprehend directly (the Earth, which is vast and beyond experiencing when trapped on the ground) and describing it as something we can hold and feel and see, easily. But if we are using macro-models of our own experiences with which to describe other things also in our experience, then....

Well, what if our senses of nature are lying to us?? What if what our senses tell us and the general models that we imagine are present as a result of these senses are wrong? Consistent, perhaps. But consistently wrong? How do we escape this problem?

We know for a fact that our senses do "lie" to us. For example, it is obvious from some of our sensations that the Earth does not move. Our sense of motion works pretty good when we are walking, floating in water, running, etc. Yet when we just sit in one spot and try to feel motion, we feel none. But the Earth does move.

What causes things to sink or float? Our senses are completely lacking a density detector. We can feel weight. We can feel bulk. But we have nothing to inform us well about the ratio of mass to volume. This is a scientific concept, developed and derived after much investigation and effort, to help us understand why things "sink or float." But we have no direct sensation of it and, more truly, cannot even invent or recognize the idea of density without a theory about it (as history shows us well.)

So what do we do? We cannot trust our direct sensations. We cannot even trust that the models of thinking that we develop as a result of these questionable sensations are useful for anything more than a shallow understanding. How do we delve deeply?

How I personally like to "see" or imagine mathematics applying into physics (describing and predicting our natural universe that relates to our perceptions of it) is this way:

Imagine a man who has long thought on the nature of things and realizes the above quandry. Wanting to find some way out, some way to escape what his senses try to convince him of, to think clearly in some fashion... he decided to try something new. He goes into a deep hole, seals up his eyes, covers his other senses, and covers over himself to try and block out all that the world screams into his ears and shines into his eyes. He tries to exclude the world. To get rid of it so that he can think without the distractions and try to gain some clearer view.

For a thousand years he stays. A thousand years of blocking out his senses, leaving only his mind as his resource. In the process, he develops the idea of a point, a line, a plane. None of which have anything really to do with nature or his sensations of it. Abstract ideas, instead. He pursues the implications of just a few basic assumptions whereever that takes his mind. Again, none of this connects with nature, nor is confused by experiencing it. He rigorously develops whole tapestries of ideas, investigates various avenues and their implications, and discovers a new world within his mind. A precise world and highly unified world. But one that is unique and apart from ordinary human experience and sensation.

A thousand years in the hole. A thousand years thinking about things unrelated to nature. A thousand years of developing rigous ideas that can be communicated precisely to anyone, whether they be in that day and age or another, whether they be black or white, and no matter what the fad is or the ambient paradigms of the culture. And where if one person sees a deduction, another will see the same deduction.

Finally, after all this time, he opens his eyes; exposes his skin; breathes in the air once again; and looks upon the world once more.

But with a new inner "eye". A new set of tools upon which to judge this world, to measure it -- with which to observe it. And no longer will his examinations be tricked by his own sensations and models developed solely from them.

It's like that, in a sense. There is much more to be filled in here and I've left gaping holes. But the basic image is there.

Jon

rigorous

Oh, well! So much for my once-perfect mind!

Jon

Or at least you thought it was! ;-)

By the way, this series appears to be... old, not reviewed much, and perhaps in need of at least some editing.

I was just perusing quickly through module 2 and early on found two things that make me wonder about it. The first was the first sentence on page 1-4 saying, "The sine wave illustrated in figure 1-1 view B is a plot of a current which changes amplitude and direction." That figure is at the bottom of page 1-3 and it shows a y-axis scale going from +V to -V. A voltage, not a "current." The second was a comment at the top of page 1-12 which says, "Recently the HERTZ (Hz) has been designated to indicate one cycle per second." I'm sure that by some measure it was recent. But that isn't the kind of phrasing one usually gets from modern texts, especially as the term was set up by the IEC in 1930, certainly in some use in the 50's if I recall, and I think it got adopted by the CGPM (whose French name I will never remember) in 1960.

I guess some things never need to change so they left these earlier books untouched basically from WW II?

Jon

Speaking of pop cans, the metal is much thinner on the sides than it is on the top or bottom. Stick that into your optimizing algorithm and crank it.

That's a function of the drawing process which has two or more stages. You can't draw the bottom much and the top is stamped.

Excellent points about the gripping and spilling. And if they made soda cans much taller and narrower, there would be problems with cans tipping over too easily.

Mark

The sides are squished out by a high speed stamp. The real optimization problem is how thin you can make them while keeping them strong enough. But for word problem purposes you can always assign separate thicknesses to the side and the ends, and ask for the shape that minimizes the metal per unit volume.

There was an experiment somewhere, I heard about it from NIST people. They needed some thin-walled aluminum pressure vessels. Nobody does that better than the beer companies. Coors donated some empty and unlabeled "Silver Bullet" cans, which worked fine.

You should see the problem disabled people have holding a regular soda can. A neighbor of mine has arthritis so bad that her hands look like hooks, and she has to struggle with everything.