[Blindmath] grayscale braille

Thu Mar 22 06:31:24 UTC 2012

Well, I don't know much about souls, but let's see if this analogy makes
any sense.

I like to think of color in an image as being analogous to bandwidth in a
communication channel.

By increasing the bandwidth of a communication channel, we can convey more
information per unit of time.

By adding color to images (as compared to gray scale), we can convey more
information per square centimeter of image space.

Of course, more information is only useful if more information is needed.
And it is certainly possible for the human brain to become overwhelmed by
too much information.

So don't think about color in a visual sense. After all, color is just a
bunch of numbers. And numbers should mean the same thing to all of us
whether sighted or blind.

Instead, think about color in an image as a mathematical description of a
surface, such as a contour map of geographical area, a weather map showing
areas of high and low pressure, or the evaluation of a mathematical
expression such as the so called "Mexican hat" equation (see
http://seis.bris.ac.uk/~tm7976/human.htm)

Then ask yourself if more dynamic range in describing that surface would be
useful. If the answer is yes, then the use of color might be appropriate.
If the answer is no, then don't complicate things by adding color.

Dick Baldwin

On Wed, Mar 21, 2012 at 10:09 AM, <bente at casilenc.com> wrote:

> Hey guys,
>
>  Well I have been following this discussion and while I cannot add any
> technical perspective, I do not see a value add to focus much on color.
> I will use a photography analogy since that is a hobby of mine.  I heard
> it said once that a color picture is just that, you see the colors, but
> a black and white photo shows you the picture's soul.   Colors can be a
> distraction from "the main idea" the picture is trying to convey.  Just
> my two cents for the day.
>
> Bente
>
>
>
>
>
>
>
>  Hello,
> > Just a few quick things. Yes I understand what you say about colours in
> > diagrams, I think I probably was getting at that this colour
> > representation
> > system probably would not be good for use with pictures (eg. where it is
> > meant to be a life like representation of something). I guess it relates
> > to
> > a feeling I have towards many of these artificial sight implants, why
> > would
> > I want one if I could not actually appreciate visual things with it (eg.
> > could I appreciate a painting). Any other use of it normally is getting
> > information and well do I really need it in a "sight form"? As an example
> > I
> > remember hearing a thing in the news where it said someone with such an
> > implant could see to tell the time, well I can do that just as well with
> a
> > much less expensive (approx £12) Braille watch.
> >
> > Well where am I going with this? Not quite sure but I think its to do
> with
> > does one then really need to know the actual colour composition in a
> > diagram
> > representation. Probably not, and therefore does one only need to know
> > which
> > of 27 (or how ever many you wish to choose) combinations is in a given
> > pixel. I really couldn't put a name to 66% red, 66% green and 33% blue,
> > just
> > knowing its colour 13 would be enough for identification purposes.
> >
> > Funny you mention cartoons in your examples of recognising things. When I
> > was younger I did see The Simpsons, and always found Bart Simpson an
> > interesting character for a long time as they used the same colour for
> his
> > skin and hair and with the spiked up hair it made him look like he had a
> > bald head with a jagged edge. I say it for interest purposes, it just
> adds
> > to what you say as if they had used a different colour for hair and skin
> > such an affect would not appear.
> >
> > Michael Whapples
> >
> > -----Original Message-----
> > From: Richard Baldwin
> > Sent: Wednesday, March 21, 2012 2:16 PM
> > To: Blind Math list for those interested in mathematics
> > Subject: Re: [Blindmath] grayscale braille
> >
> > Hi Michael,
> >
> > You asked:
> >
> > "Out of interest, having discussed how colours are represented, what was
> > 256 colours about on some of the early windows computers, 256 is not
> > anything raised to the three."
> >
> > As it turns out, the mechanism for dealing with colors today is less
> > complex than was the case in the early days of computers when memory and
> > file space was more expensive.
> >
> > As you are aware, 256 is the number of possible combinations of eight
> bits
> > in a standard 8-bit byte of memory or file space. I don't remember all of
> > the details, but in some fashion, the color data was arranged so that one
> > color was assigned to each combination of bits. This allowed the maximum
> > number of colors to be used in an image with only one byte per pixel,
> thus
> > minimizing the storage requirements for images. There were other schemes
> > in
> > use as well that allowed for a smaller set of colors to be used, but
> > allowed the user (or the programmer) to specify the colors in the set.
> >
> > Fortunately, it is no longer necessary to deal with that sort of
> > complexity. Color is typically represented with three 8-bit primary color
> > bytes and an alpha byte.  The alpha byte deals with how a new color
> blends
> > with an old color when the new color is painted on top of an existing
> > color. The four bytes fit nicely in a 32-bit int value.
> >
> > However, it is still possible to "personalize" a Windows desktop such
> that
> > color is represented in 16 total bits. I don't know how those 16 bits are
> > allocated among the color values, but I suspect that the alpha value
> > doesn't exist in the 16-bit format.
> >
> > You also wrote:
> >
> > "So I guess my thought is simply, colour in tactile diagrams will only be
> > useful for area identification rather than to convey how something
> > visually
> > appears.
> >
> > I understand that and I believe that for the most part, it doesn't
> matter.
> > To some extent that is true for sighted people as well. Let me illustrate
> > with an anecdotal example. My wife and I elected not to purchase a color
> > TV
> > for many years after they became readily available. During the many years
> > of watching "black and white" TV, I saw a particular commercial for a
> > hamburger chain hundreds of times. Although I never thought about it, I
> > had
> > in my mind that the uniforms worn by the people who worked there were
> > different shades of blue and they wore dark blue hats.
> >
> > Then one day I stopped at one of the stores to purchase a hamburger and
> > was
> > shocked to learn that the uniforms were actually red and orange. However,
> > the fact that the employees wore red hats instead of blue hats wasn't
> > important in my ability to extract information from the image. I was
> still
> > able to distinguish between the hat and the face on the basis of color
> > differences.
> >
> > In my mind, unless an image is being used to teach a blind student about
> > colors in the real world, the absolute values of the individual colors in
> > the image are not terribly important. It is the differences in the colors
> > that conveys information, not the absolute colors.
> >
> > The importance of colors in an image is in distinguishing the features of
> > an image from one another. For example, a plot of the function y=sin(x)
> > means the same thing with a blue line on a pink background as it does
> with
> > a black line on a white background. In either case, the line represents
> > the
> > values of the function.
> >
> > Blue on pink might seem a little strange to a sighted person because we
> > have been mentally conditioned through past experience to expect black on
> > white. However, it is the difference in colors that conveys the
> > information
> > and not the absolute values of the colors themselves. Had it been most
> > economical to produce trigonometry textbooks using blue on pink in the
> > past, that would be the norm and black on white might look strange.
> >
> > Another example that minimizes the importance of absolute colors is the
> > drawing of cartoons. Many cartoons characters have facial features that
> > are
> > distinguished by non real-world colors (such as purple faces). That
> > doesn't
> > stop children from recognizing those features as noses, eyes, chins, etc.
> >
> > It is computationally very easy to mathematically invert colors.
> Inverting
> > all of the colors in an image changes the absolute values of the colors
> > while maintaining the differences between colors.
> >
> > Unless the subtle details of an image depend on the use of real-world
> > colors, (such as distinguishing between apples and oranges in a bowl of
> > fruit), I can invert all of the colors in an image and a sighted observer
> > of the image will still be able to comprehend the information content of
> > the image.
> >
> > Bottom line: color is simply a mathematical concept consisting of
> specific
> > arrangements of numeric values. It is usually the differences in the
> > colors
> > that conveys information and not the absolute values of the colors
> > themselves.
> >
> > While it might be nice for blind students to be able to relate absolute
> > color values to the real world, in general, that is not a requirement for
> > extracting and understanding the information content of an image. The
> > information content of the image is encapsulated in the changes in color
> > values and not in the absolute values of the colors.
> >
> > More food for thought.
> >
> > Dick Baldwin
> >
> > On Wed, Mar 21, 2012 at 7:20 AM, Michael Whapples <mwhapples at aim.com>
> > wrote:
> >
> >> Thanks for that, it just sparked of a logical way of assigning in my
> >> mind.
> >>
> >> However before discussing the system, may be quick thoughts on colours
> >> and
> >> how someone who cannot see relates to them. I have seen a little in the
> >> past so do understand what colours are and can imagine what a colour
> >> might
> >> be like if someone can start from a point I know and then describe how
> >> it
> >> varies (eg. darker, a bit greener, etc). However if I don't have a
> >> decent
> >> start point close to where we need to get I can actually struggle a bit
> >> to
> >> understand what it might look like (eg. I am getting an artificial eye,
> >> I
> >> get to choose my eye colour how great, however when discussing it I was
> >> struggling at times to know what colour precisely we were discussing and
> >> whether it suits me as I have never seen well enough to pick out eye
> >> colours). So I guess my thought is simply, colour in tactile diagrams
> >> will
> >> only be useful for area identification rather than to convey how
> >> something
> >> visually appears.
> >>
> >> Back to how to show colour in a tactile form. We will stick with your 27
> >> levels to start with. Each row of the Braille cell (6 dot cell) could
> >> represent each colour, and the number of dots on that row tells you the
> >> level for the colour. Lets just assign the rows logically, RGB, top,
> >> middle
> >> bottom rows respectively.
> >>
> >> Now the above also fits with your previous suggestion of filling up the
> >> cell means more intense and the overall view of the cell relates to
> >> that.
> >>
> >> However may be at the cost of loosing that relationship of more dots
> >> more
> >> intense, we could actually extend up to 64 colours by using the position
> >> of
> >> just one dot on a row (left or right side of the cell) so allowing each
> >> colour to have four intensities.
> >>
> >> Whether 27 or 64 is needed I don't know, in either case its still an
> >> improvement over what sighted people had with computers back when I
> >> started
> >> with a BBC micro (8 or 16 depending on what computer system you went
> >> with).
> >>
> >> Out of interest, having discussed how colours are represented, what was
> >> 256 colours about on some of the early windows computers, 256 is not
> >> anything raised to the three.
> >>
> >>
> >> Michael Whapples
> >>
> >> -----Original Message----- From: Richard Baldwin
> >> Sent: Wednesday, March 21, 2012 12:57 AM
> >>
> >> To: Blind Math list for those interested in mathematics
> >> Subject: Re: [Blindmath] grayscale braille
> >>
> >> I'm going to add a few more comments to those that I made earlier. As
> >> you
> >> may have gathered, I have had a long career in computer technology. That
> >> career included a large dose of digital signal processing. I will get to
> >> the reason for that comment later.
> >>
> >> Once we learn of a new technology, it is interesting to speculate on how
> >> that technology might be put to use. John indicated earlier that his
> >> ASCII
> >> Gray Scale technology will provide 26 levels of gray scale. Since I know
> >> little to nothing about Braille, I will take him at his word on that.
> >>
> >> My digital signal processing activities included working for several
> >> years
> >> in the area of undersea sonar technology. At that time, the general
> >> consensus was that having only a gray scale display, an experienced
> >> sonar
> >> operator could only make use of about seven levels of gray. Thus, while
> >> 26
> >> levels of gray might be useful for some applications, it might be
> >> overkill
> >> for others.
> >>
> >> It was also the general consensus that an experienced sonar operator
> >> using
> >> a color display could make use of far more than 7 different colors. This
> >> caused me to wonder if it might make sense to apply those 26 levels to
> >> color as an alternative to gray scale. I wonder if 26 levels of color
> >> information might be more useful than 26 levels of gray scale
> >> information.
> >>
> >> I am going to assume that such a possibility might be worth looking into
> >> and for reasons that I will explain later, I am going to assume that by
> >> including one non-alphabetic character, it would be possible to expand
> >> to
> >> 27 levels instead of 26.
> >>
> >> A typical full-color bitmap image contains more than 16 million colors.
> >> Therefore, converting those colors to either 26 gray scale values or 27
> >> color values will necessarily cause a large amount of information to be
> >> lost.
> >>
> >> However, in many cases, converting to 26 gray scale levels will cause
> >> much
> >> more information to be lost than converting to 26 or 27 colors.
> >>
> >> Converting to gray scale means throwing away the color information and
> >> preserving only intensity information without making any attempt to make
> >> use of the color information.
> >>
> >> Reducing the number of colors down to 27 preserves both color
> >> information
> >> and intensity information and may produce a more useful result.
> >>
> >> Having decided that it might be useful to re-quantize an image into 27
> >> unique colors leaves open the question of how best to do that. There are
> >> a
> >> multitude of possibilities in this regard and some may prove more useful
> >> than others. I will describe one such approach below.
> >>
> >> First some background information. Color in a modern computer is
> >> typically
> >> maintained as a weighted sum of three primary colors: red, green, and
> >> blue.
> >>
> >> Displaying those three primary colors along with three secondary colors
> >> produced by adding the three primary colors together in pairs produces
> >> the
> >> six colors of a typical rainbow: red, yellow, green, cyan, blue, and
> >> magenta.
> >>
> >> While this RGB approach to dealing with color is particularly efficient
> >> for
> >> computational purposes, it does not describe how humans think about
> >> color.
> >> For example, it is not likely that a human would go to a furniture store
> >> and ask to see a sofa with upholstery that is X-percent red, Y-percent
> >> green, and Z-percent blue. Instead, in Austin, Texas, where burnt orange
> >> is
> >> the color of the day, (at least for UT students) a UT student might ask
> >> for
> >> a color that is a subdued version of orange that is a little on the dark
> >> side and closer to red than green, or words to that effect.
> >>
> >> Another system for dealing with color that is more in keeping with how
> >> humans think about color is often called HSV or HSB. This stands for
> >> Hue,
> >> Saturation, and either Value or Brightness depending on the use of a V
> >> or
> >> a
> >> B in the abbreviation.
> >>
> >> Fortunately, it is relatively easy to write a computer program that will
> >> transform a set of unique RGB color values to corresponding unique HSB
> >> values and back again.
> >>
> >> In the RGB system, the values of red, green, and blue for any individual
> >> pixel can range from 0 to 255. Thus, the number of possible combinations
> >> is
> >> equal to 255 raised to the third power, which is where the "more than 16
> >> million" comes from. In fact, a typical modern computer system using
> >> 24-bit
> >> RGB color can mathematically describe 16,581,375 different colors.
> >>
> >> Similarly, these 16 million plus RGB colors can be transformed into a
> >> corresponding set of 16 million unique colors in the HSB nomenclature.
> >> In
> >> the HSB nomenclature, the values for each of the three parameters ranges
> >> from 0 to 1.0.
> >>
> >> One approach to converting from more than 16 million unique colors to 27
> >> unique colors would be to:
> >>
> >> 1. Transform the RGB color for each pixel to the corresponding HSB
> >> color.
> >> 2. Re-quantize each of the three HSB parameter values into three unique
> >> values: 0.333, 0.666, and 1.0. This would produce 3-cubed or 27 unique
> >> colors.
> >> 3. Transform the modified HSB values back to the corresponding RGB
> >> values
> >> and replace the original pixel color with the new color.
> >>
> >> Given the 27 possible color values, a table lookup procedure could be
> >> used
> >> to generate an ASCII value corresponding to each color value and that
> >> ASCII
> >> value could be used to produce the Braille characters.
> >>
> >> Food for thought.
> >>
> >> Dick Baldwin
> >>
> >>
> >>
> >> On Tue, Mar 20, 2012 at 1:42 PM, Richard Baldwin
> >> <baldwin at dickbaldwin.com>
> >> **wrote:
> >>
> >>  Hi Michael,
> >>>
> >>> You wrote:
> >>>
> >>> "may not really give the same impression as the visual things you
> >>> describe as one never reads/views Braille at a distance so won't get
> >>> that
> >>> point of not really resolving individual dots but rather a general
> >>> effect.
> >>> "
> >>>
> >>> In this respect, I was thinking in terms of embossed images as opposed
> >>> to
> >>> single line Braille displays. It seems to me that brushing your hand
> >>> across
> >>> an embossed image and being aware of the dot density at different
> >>> locations
> >>> on the image might be somewhat analogous to viewing a printer-art image
> >>> from a distance.
> >>>
> >>> Dick B.
> >>>
> >>> On Tue, Mar 20, 2012 at 1:26 PM, Michael Whapples <mwhapples at aim.com
> >>> >wrote:
> >>>
> >>>  Wow, what a lot of information.
> >>>>
> >>>> I hadn't heard of the making a grayscale image with people before, I
> >>>> knew
> >>>> people have used lots of people to create images by having people
> >>>> wearing
> >>>> different colours though.
> >>>>
> >>>> Back to the actual subject, I thought you might have been interested
> >>>> as
> >>>> I
> >>>> know with some of your image processing you have commented on how low
> >>>> the
> >>>> grayscale resolution (shades of gray resolution) is with existing
> >>>> technology. However the increase in gray levels comes at a very high
> >>>> cost
> >>>> of spatial resolution, is 40 pixels wide, may be 80 for very
> >>>> rich/lucky
> >>>> people with such a Braille display, really good enough? I have my
> >>>> doubts
> >>>> even for those with 80 cell displays, the display is so long will one
> >>>> really get a proper awareness of what relates to what because of the
> >>>> spread?
> >>>>
> >>>> Another question is, why only use letters? Surely one could
> >>>> potentially
> >>>> use up to 64 for a 6-dot cell and on a Braille display one could get
> >>>> full
> >>>> 256 level representation!
> >>>>
> >>>> I like your thought of number of dots for level as it would make it
> >>>> intuitive, but does then bring down the number of levels and may not
> >>>> really
> >>>> give the same impression as the visual things you describe as one
> >>>> never
> >>>> reads/views Braille at a distance so won't get that point of not
> >>>> really
> >>>> resolving individual dots but rather a general effect.
> >>>>
> >>>> To try to keep to some sort of logical assignment, if trying for the
> >>>> 256
> >>>> levels, then I would just use binary around the Braille cell (eg.
> >>>> lightest
> >>>> being no dots, next coming dot-1, next being dot-2, next being
> >>>> dots-12,...
> >>>> very nearly black dots-2345678, darkest dots-12345678).
> >>>>
> >>>> Michael Whapples
> >>>>
> >>>> -----Original Message----- From: Richard Baldwin
> >>>> Sent: Tuesday, March 20, 2012 4:37 PM
> >>>>
> >>>> To: Blind Math list for those interested in mathematics
> >>>> Subject: Re: [Blindmath] grayscale braille
> >>>>
> >>>> I have been following this conversation with interest. For the record,
> >>>> I
> >>>> am
> >>>> not blind and know very little about Braille. However, I do know quite
> >>>> a
> >>>> lot about image processing.
> >>>>
> >>>> In the sighted world, a character printer can definitely be used to
> >>>> produce
> >>>> gray scale images, but not in the way that is described here.
> >>>>
> >>>> In the 1960s, a typical data processing printer weighed several
> >>>> hundred
> >>>> pounds, stood chest high from the floor, printed upper-case letters,
> >>>> numbers, and a set of special characters at 10 characters per inch
> >>>> with
> >>>> a
> >>>> line length of 132 characters on 14-inch wide fan-fold paper. A
> >>>> typical
> >>>> data processing printer could print 600 lines per minute or more. With
> >>>> some
> >>>> printers, the paper came out so fast that special mechanisms were
> >>>> required
> >>>> to prevent it from flying across the room and to refold itself in the
> >>>> output bin.
> >>>>
> >>>> Many data centers had various examples of printer art posted on the
> >>>> walls
> >>>> with the most common being a reasonably good gray scale replica of the
> >>>> Mona
> >>>> Lisa.
> >>>>
> >>>> However, unlike the scheme that is described here, there was no intent
> >>>> for
> >>>> the viewer to assign special meaning to any individual character. In
> >>>> fact,
> >>>> the intent was for the characters to visually run together is such a
> >>>> way
> >>>> that they would not be perceived as characters at all. Instead, the
> >>>> big
> >>>> picture view of the printout would give the impression of a gray scale
> >>>> image with individual characters fading into the background.
> >>>>
> >>>> Someone came up with a sequence of characters based on the amount of
> >>>> ink
> >>>> deposited within the 0.1-inch wide cell by each character. That was a
> >>>> long
> >>>> time ago and I don't recall the specific sequence of characters that
> >>>> was
> >>>> used. I am guessing that the period character was used to convey light
> >>>> gray. Moving from there through the sequence, each character deposited
> >>>> more
> >>>> ink and therefore produced a darker cell. I'm also guessing that the
> >>>> sequence probably consisted of eight to ten different characters
> >>>> making
> >>>> it
> >>>> possible to produce the illusion of eight to ten levels of gray.
> >>>>
> >>>> Characters were chosen such that when a person stepped away from the
> >>>> printout and viewed it as a whole, that person didn't see individual
> >>>> characters. Instead, the result was an illusion of a large gray scale
> >>>> image.
> >>>>
> >>>> I used this scheme myself in the days before the invention of the
> >>>> CalComp
> >>>> incremental plotter to produce images of contour maps.
> >>>>
> >>>> Perhaps a similar scheme could be used with braille with each cell
> >>>> containing from zero to six dots (or perhaps eight dots). This might
> >>>> make
> >>>> it possible for a blind person to perceive white plus six (or eight)
> >>>> levels
> >>>> of gray without the requirement to mentally associate specific
> >>>> characters
> >>>> with specific shades of gray.
> >>>>
> >>>> For the six-dot case, the following sequence of characters might
> >>>> provide
> >>>> the illusion of increasing darkness (but a different selection might
> >>>> produce better tactile results):
> >>>>
> >>>> hex 41, A, 1 dot
> >>>> hex 42, B, 2 dots
> >>>> hex 44, D, 3 dots
> >>>> hex 47, G, 4 dots
> >>>> hex 51, Q, 5 dots
> >>>> hex 3D, =, 6 dots
> >>>>
> >>>> Dots have long been used to produce the illusion of gray scale images.
> >>>> When
> >>>> I was a youngster, pictures in most small-town newspapers were
> >>>> presented
> >>>> in
> >>>> gray scale because printing presses that could print in color were
> >>>> very
> >>>> expensive. If you looked closely at a newspaper photo, you could see
> >>>> that
> >>>> the picture was simply an array of dots. I seem to recall that the
> >>>> gray
> >>>> scale effect was achieved by producing an array of black dots on a
> >>>> uniform
> >>>> grid using different sized dots.
> >>>>
> >>>> At one point in time, I had some very interesting photographs from
> >>>> "Life"
> >>>> magazine involving very unique gray scale images. In those photos, a
> >>>> photographer produced images of various things, including a portrait
> >>>> of
> >>>> Woodrow Wilson and a picture of the U.S. Marine insignia by taking
> >>>> photographs of thousands of troops in formation wearing white shirts
> >>>> and
> >>>> black shirts. In effect, each person was one dot in the image.
> >>>>
> >>>> Apparently the photographer would place the camera on top of a
> >>>> building
> >>>> or
> >>>> tower and take of picture of the troops in formation down below. He
> >>>> even
> >>>> took perspective into account. For example, moving away from the
> >>>> camera,
> >>>> each row of troops was wider than the one before it. In some cases,
> >>>> the
> >>>> row
> >>>> of troops closest to the camera contained 20 or 30 troops while the
> >>>> most
> >>>> distant row would contain 200 to 300 troops. Thus, the "dot density"
> >>>> increased as you viewed the image going from bottom to top.
> >>>>
> >>>> I was able to find an image of a printer generated Mona Lisa on the
> >>>> web,
> >>>> but was unable to find any images of the human-dot photographs.
> >>>>
> >>>> Dick Baldwin
> >>>>
> >>>>
> >>>> On Mon, Mar 19, 2012 at 6:13 PM, Michael Whapples <mwhapples at aim.com>
> >>>> wrote:
> >>>>
> >>>>  I don't know if John is on the blindmath list, I'll forward the
> >>>> message
> >>>>
> >>>>> on
> >>>>> in case he isn't, however it would be better if you could somehow
> >>>>> communicate direct with him (eg. by posting to the NFB-science list
> >>>>> if
> >>>>> he
> >>>>> isn't on blindmath).
> >>>>>
> >>>>> Michael Whapples
> >>>>>
> >>>>> -----Original Message----- From: Pranav Lal
> >>>>> Sent: Monday, March 19, 2012 10:58 PM
> >>>>> To: 'Blind Math list for those interested in mathematics'
> >>>>> Subject: Re: [Blindmath] grayscale braille
> >>>>>
> >>>>>
> >>>>> Hi John,
> >>>>>
> >>>>> Can I use this plotting technique for any image? Your example seems
> >>>>> to
> >>>>> work
> >>>>> only for functions.
> >>>>>
> >>>>> Pranav
> >>>>>
> >>>>>
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> >>>>>
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> >>>>> Blindmath:
> >>>>> http://nfbnet.org/mailman/******options/blindmath_nfbnet.org/******<
> http://nfbnet.org/mailman/****options/blindmath_nfbnet.org/****>
> >>>>> <http://nfbnet.org/mailman/****options/blindmath_nfbnet.**org/**<
> http://nfbnet.org/mailman/**options/blindmath_nfbnet.org/**>
> >>>>> >
> >>>>> mwhapples%40aim.com<http://**n**fbnet.org/mailman/options/**<
> http://nfbnet.org/mailman/options/**>
> >>>>> blindmath_nfbnet.org/****mwhapples%40aim.com<
> http://blindmath_nfbnet.org/**mwhapples%40aim.com>
> >>>>> <http://**nfbnet.org/mailman/options/**blindmath_nfbnet.org/**
> >>>>> mwhapples%40aim.com<
> http://nfbnet.org/mailman/options/blindmath_nfbnet.org/mwhapples%40aim.com
> >
> >>>>> >
> >>>>> >
> >>>>>
> >>>>> ______________________________******_________________
> >>>>> Blindmath mailing list
> >>>>> Blindmath at nfbnet.org
> >>>>> http://nfbnet.org/mailman/******listinfo/blindmath_nfbnet.org<
> http://nfbnet.org/mailman/****listinfo/blindmath_nfbnet.org>
> >>>>> <**http://nfbnet.org/mailman/****listinfo/blindmath_nfbnet.org<
> http://nfbnet.org/mailman/**listinfo/blindmath_nfbnet.org>
> >>>>> >
> >>>>> <**http://nfbnet.org/mailman/****listinfo/blindmath_nfbnet.org<
> http://nfbnet.org/mailman/**listinfo/blindmath_nfbnet.org>
> >>>>> **<http://nfbnet.org/mailman/**listinfo/blindmath_nfbnet.org<
> http://nfbnet.org/mailman/listinfo/blindmath_nfbnet.org>
> >>>>> >
> >>>>>
> >>>>> >
> >>>>>
> >>>>> To unsubscribe, change your list options or get your account info for
> >>>>> Blindmath:
> >>>>> http://nfbnet.org/mailman/******options/blindmath_nfbnet.org/******<
> http://nfbnet.org/mailman/****options/blindmath_nfbnet.org/****>
> >>>>> <http://nfbnet.org/mailman/****options/blindmath_nfbnet.**org/**<
> http://nfbnet.org/mailman/**options/blindmath_nfbnet.org/**>
> >>>>> >
> >>>>> baldwin%40dickbaldwin.com<**http**://nfbnet.org/mailman/**options/**
> <http://nfbnet.org/mailman/options/**>
> >>>>> blindmath_nfbnet.org/baldwin%****40dickbaldwin.com<http://**
> >>>>> nfbnet.org/mailman/options/**blindmath_nfbnet.org/baldwin%**
> >>>>> 40dickbaldwin.com<
> http://nfbnet.org/mailman/options/blindmath_nfbnet.org/baldwin%40dickbaldwin.com
> >
> >>>>> >
> >>>>> >
> >>>>>
> >>>>>
> >>>>>
> >>>>
> >>>> --
> >>>> Richard G. Baldwin (Dick Baldwin)
> >>>> Home of Baldwin's on-line Java Tutorials
> >>>> http://www.DickBaldwin.com
> >>>>
> >>>> Professor of Computer Information Technology
> >>>> Austin Community College
> >>>> (512) 223-4758
> >>>> mailto:Baldwin at DickBaldwin.com
> >>>> http://www.austincc.edu/****baldwin/<
> http://www.austincc.edu/**baldwin/><
> >>>> http://www.austincc.edu/**baldwin/ <http://www.austincc.edu/baldwin/
> >>
> >>>>
> >>>> ______________________________****_________________
> >>>> Blindmath mailing list
> >>>> Blindmath at nfbnet.org
> >>>> http://nfbnet.org/mailman/****listinfo/blindmath_nfbnet.org<
> http://nfbnet.org/mailman/**listinfo/blindmath_nfbnet.org>
> >>>> <**http://nfbnet.org/mailman/**listinfo/blindmath_nfbnet.org<
> http://nfbnet.org/mailman/listinfo/blindmath_nfbnet.org>
> >>>> >
> >>>> To unsubscribe, change your list options or get your account info for
> >>>> Blindmath:
> >>>> http://nfbnet.org/mailman/****options/blindmath_nfbnet.org/****<
> http://nfbnet.org/mailman/**options/blindmath_nfbnet.org/**>
> >>>> mwhapples%40aim.com<http://**nfbnet.org/mailman/options/**
> >>>> blindmath_nfbnet.org/**mwhapples%40aim.com<
> http://nfbnet.org/mailman/options/blindmath_nfbnet.org/mwhapples%40aim.com
> >
> >>>> >
> >>>>
> >>>> ______________________________****_________________
> >>>> Blindmath mailing list
> >>>> Blindmath at nfbnet.org
> >>>> http://nfbnet.org/mailman/****listinfo/blindmath_nfbnet.org<
> http://nfbnet.org/mailman/**listinfo/blindmath_nfbnet.org>
> >>>> <**http://nfbnet.org/mailman/**listinfo/blindmath_nfbnet.org<
> http://nfbnet.org/mailman/listinfo/blindmath_nfbnet.org>
> >>>> >
> >>>> To unsubscribe, change your list options or get your account info for
> >>>> Blindmath:
> >>>> http://nfbnet.org/mailman/****options/blindmath_nfbnet.org/****<
> http://nfbnet.org/mailman/**options/blindmath_nfbnet.org/**>
> >>>> baldwin%40dickbaldwin.com<http**://nfbnet.org/mailman/options/**
> >>>> blindmath_nfbnet.org/baldwin%**40dickbaldwin.com<
> http://nfbnet.org/mailman/options/blindmath_nfbnet.org/baldwin%40dickbaldwin.com
> >
> >>>> >
> >>>>
> >>>>
> >>>
> >>>
> >>> --
> >>> Richard G. Baldwin (Dick Baldwin)
> >>> Home of Baldwin's on-line Java Tutorials
> >>> http://www.DickBaldwin.com
> >>>
> >>> Professor of Computer Information Technology
> >>> Austin Community College
> >>> (512) 223-4758
> >>> mailto:Baldwin at DickBaldwin.com
> >>> http://www.austincc.edu/**baldwin/ <http://www.austincc.edu/baldwin/>
> >>>
> >>>
> >>
> >>
> >> --
> >> Richard G. Baldwin (Dick Baldwin)
> >> Home of Baldwin's on-line Java Tutorials
> >> http://www.DickBaldwin.com
> >>
> >> Professor of Computer Information Technology
> >> Austin Community College
> >> (512) 223-4758
> >> mailto:Baldwin at DickBaldwin.com
> >> http://www.austincc.edu/**baldwin/ <http://www.austincc.edu/baldwin/>
> >> ______________________________**_________________
> >> Blindmath mailing list
> >> Blindmath at nfbnet.org
> >> http://nfbnet.org/mailman/**listinfo/blindmath_nfbnet.org<
> http://nfbnet.org/mailman/listinfo/blindmath_nfbnet.org>
> >> To unsubscribe, change your list options or get your account info for
> >> Blindmath:
> >> http://nfbnet.org/mailman/**options/blindmath_nfbnet.org/**
> >> mwhapples%40aim.com<
> http://nfbnet.org/mailman/options/blindmath_nfbnet.org/mwhapples%40aim.com
> >
> >>
> >> ______________________________**_________________
> >> Blindmath mailing list
> >> Blindmath at nfbnet.org
> >> http://nfbnet.org/mailman/**listinfo/blindmath_nfbnet.org<
> http://nfbnet.org/mailman/listinfo/blindmath_nfbnet.org>
> >> To unsubscribe, change your list options or get your account info for
> >> Blindmath:
> >> http://nfbnet.org/mailman/**options/blindmath_nfbnet.org/**
> >> baldwin%40dickbaldwin.com<
> http://nfbnet.org/mailman/options/blindmath_nfbnet.org/baldwin%40dickbaldwin.com
> >
> >>
> >
> >
> >
> > --
> > Richard G. Baldwin (Dick Baldwin)
> > Home of Baldwin's on-line Java Tutorials
> > http://www.DickBaldwin.com
> >
> > Professor of Computer Information Technology
> > Austin Community College
> > (512) 223-4758
> > mailto:Baldwin at DickBaldwin.com
> > http://www.austincc.edu/baldwin/
> > _______________________________________________
> > Blindmath mailing list
> > Blindmath at nfbnet.org
> > http://nfbnet.org/mailman/listinfo/blindmath_nfbnet.org
> > To unsubscribe, change your list options or get your account info for
> > Blindmath:
> >
> http://nfbnet.org/mailman/options/blindmath_nfbnet.org/mwhapples%40aim.com
> >
> >
> > _______________________________________________
> > Blindmath mailing list
> > Blindmath at nfbnet.org
> > http://nfbnet.org/mailman/listinfo/blindmath_nfbnet.org
> > To unsubscribe, change your list options or get your account info for
> > Blindmath:
> >
> http://nfbnet.org/mailman/options/blindmath_nfbnet.org/bente%40casilenc.com
> >
>
>
>
> _______________________________________________
> Blindmath mailing list
> Blindmath at nfbnet.org
> http://nfbnet.org/mailman/listinfo/blindmath_nfbnet.org
> To unsubscribe, change your list options or get your account info for
> Blindmath:
>
> http://nfbnet.org/mailman/options/blindmath_nfbnet.org/baldwin%40dickbaldwin.com
>

-- 
Richard G. Baldwin (Dick Baldwin)
Home of Baldwin's on-line Java Tutorials
http://www.DickBaldwin.com

Professor of Computer Information Technology
Austin Community College
(512) 223-4758
mailto:Baldwin at DickBaldwin.com
http://www.austincc.edu/baldwin/