[Blindmath] grayscale braille

Wed Mar 21 15:09:44 UTC 2012

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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>>>>
>>>> --
>>>> 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/>>
>>>>
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>>>> 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
>