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Going through engineering school wasn't a lot of fun for me, especially when it came to the less "obvious" maths, such as vector calculus. I've had no problem with the math I've ever needed in order to design something, but when it came to stuff like vector calc, if I didn't see an implementation, I
just couldn't do well at it.
In order to help visualize 3d vectors, I dreamed up a true 3-d screen made especially for calculators.
Current "3d" implementations rely on isometric views, and the ability to pan and rotate to allow a person to grasp a 3-d concept while drawn in 2d. This is especially difficult to manipulate with a calculator's interface, not to mention the small screen with low resolution does not lend itself well to all of this. Sure, get maple or mathematica and you're better off, but those programs have a learning curve of their own. Perchance we can make introduction to 3d vectors easier from the get-go?
Insert the true 3-d screen. The ingredients are simple but potentially expensive: Several hundred (depending on desired resolution) small but clear-when-off LCDs, ~400 pixels square. Not any LCD will do - it has to be nearly transparent when off. This does already exist, it just isn't in every implementation. Also, the design will have to be tweaked to add depth to each pixel. Current common displays (TFT, or thin film transistor) utilize a small transistor hooked into a tiny patch of liquid crystal (or three, if it is color, which this won't initially be). In order to avoid hooking up individual wires to each and every pixel, rows and columns are made up and only when voltage comes at the pixel from both sides does it activate. Given the smaller nature of the screens in this idea, i believe that adding depth to the liquid crystal packet, and the increase in power requirement would present only a small manufacturing challenge.
Once the display is assembled, the software engineers merely need to divide the image to be displayed up by the resolution and apply each picture to each separate display.
The obvious problems (to me) with this idea are display thickness, viewing angle and clarity when off. All of these issues have improved significantly over the past few years, but i believe that the viewing angle will especially benefit from a thicker crystal pad. Obviously, the viewing angle has to be a full 360deg.
Viola! now your eigenvectors have more obvious meaning.
Polaroid (from Wikipedia)
Camera company and filter brand are the same [neutrinos_shadow, Feb 26 2008]
||But, in mathematics, 3 dimensions aren't
special. You're as likely to be dealing with
four or more dimensions as with three,
||'absolutely'. 4+ dimension matricies are common in higher level math. They even come up in engineering.
The usefulness lies in more easily learning 3d (2d is easy because it fits on paper!), which may even help the student learn 4+d more easily, as, in my opinion, the jump from 2d to 3d is more difficult than the jump from 3 to 4 and then 4+.
||Fair point. Actually, I quite like the idea of
stacked LCDs. I don't think you'll need
that many to create a sufficient depth
effect. In any case, you won't be able to
view it from the side because of refractive
index variations through the different
||I have a feeling that the visibility of a pixel through the side of a screen can be made possible if it were desired. If you think of LCDs as just sandwiches of glass with liquid crystal in pockets (crystal that is clear, at least from the front, when it is not on), there seems to me to be no reason these things can't be made fairly clear edgewise.
||There is a fundamental problem with
sideways viewing, I think. Liquid crystals
work (if I remember) by twisting or not
twisting the polarization of light. For this
to work, you have to view the LCD at
roughly the right angle, so you see light
that has passed through the polaroid
||Polaroid layers, eh? (this is gonna look dumb when he fixes that spelling mistake)
Indeed TN (twisted nematic) screens rely on polarization to produce an "on" pixel, and these are what is used in your new inexpensive computer monitor.
If we choose a different type of LCD principle, we are more likely (though it is not well documented) to be able to view from the side. My first guess would be vertical alignment technology, which is actually dark when unpowered, and clear when powered. I see no reason for this to be opaque from the side.
OLEDS, while more expensive, actually emit light when electrocuted, and as such. do not require a backlight. OLEDs have a relatively shorter lifetime, between 5000 and 200,000 hours.
You do make a good point that this will not work with the cheapest of current LCD technology. However, I banish you forever to the world of the ti-30S and its 2-d ways.
||//Polaroid layers, eh?// Yes, polaroid
layers. Polaroid is the name of a type of
polarizing filter. It happens also to be the
brand name of a camera company, but
that is neither here nor there.
||I do not see much about Polaroid layers unless i search fairly specifically. Is it as common as your text would suggest?
||I suppose it depends on your background.
I've always thought it was odd that a
camera brand has the same name as a
type of polarizing film.
||//odd that a camera brand has the same name as a type of polarizing film//
According to the all-knowing Wikipedia (linky), the polarizing filters came first, and then the company expanded into sunglasses and then cameras.