[Blindmath] Pascal's Triangle in Braille?

Dave davidct1209 at gmail.com
Fri Aug 1 17:21:20 UTC 2014 

Apologies for the more than likely upcoming typo since I'm on my phone.

Suppose that you list the numbers in the following fashion in a tabular format.

1 1 1 ...
1 2 3 ...
1 3 ...
1 ...


Are the first three rows of the triangle. If you printed this table out, and just rotated the page 45° clockwise, you will have the beginnings of pascals triangle. Again, this has the obvious drawback that the numbers will be slanted in either print or Braille. You will also have to ensure that each box containing the numbers (the cells of the table) are even we spaced.

From mobile dictated

> On Aug 1, 2014, at 9:48 AM, Alexa Schriempf <ats169 at psu.edu> wrote:
> 
> Good idea. Can you say more -- I'm visualizing printing this out on our embosser. The embosser we have is an old express. As far as I know, I can't print sideways (ie portrait vs landscape orientation). So I'm not sure how I would print at 45 degrees?
>  
> perhaps I'm missing something?
>  
> thanks!
> 
> 
>> On Fri, Aug 1, 2014 at 12:42 PM, Dave via Blindmath <blindmath at nfbnet.org> wrote:
>> One of the key observations that makes the geometric or visual layout important is that the triangle is really an inverted tree.
>> 
>> Any particular note of the tree can be calculated by summing its two children
>> 
>> 
>> 
>> Here are some other ideas that you might consider.
>> If you laid out the numbers in a tabular format, and simply had this student rotate the paper 45°, you could get all of the numbers in the triangle on one sheet of paper.
>> 
>> To add even more rows, you could just attach additional papers that would fold out like a triangle, and continue the numbers that way.
>> This has the obvious drawback that the numbers will be slanted, but you can go arbitrarily deep in the tree.
>> 
>> 
>> From mobile
>> 
>> > On Jul 31, 2014, at 12:26 PM, Elise Berkley via Blindmath <blindmath at nfbnet.org> wrote:
>> >
>> > Oh, okay.  I have a better idea.  Thanks everyone.
>> >
>> > -----Original Message-----
>> > From: Blindmath [mailto:blindmath-bounces at nfbnet.org] On Behalf Of
>> > Rasmussen, Lloyd via Blindmath
>> > Sent: Thursday, July 31, 2014 11:44 AM
>> > To: 'Joseph Lee'; 'Blind Math list for those interested in mathematics'
>> > Subject: Re: [Blindmath] Pascal's Triangle in Braille?
>> >
>> > If I remember the triangle correctly, it looks somewhat like this (ignore
>> > centering in this version, it's not quite correct):
>> >                   1
>> >                1  2  1
>> >             1  3  3  1
>> >          1  4  6  4  1
>> >         1  5 10 10 5 1
>> >
>> > ... etc.
>> > I think it's easier to set this up on a brailler or a braille editor than to
>> > try to get a braille translator/formatter to do the right thing.
>> > My advice would include:  reproduce only the second half of each row
>> > (perhaps including the number before the center line to show that it is
>> > repeated).
>> > Assign the widest columns to the longest number you are going to be able to
>> > accommodate, and allow the width of your columns to get narrower as you
>> > approach the right end of a line.
>> > Omit number signs, with the possible exception of the first 2 or 3 rows.
>> > Consider producing longer rows on facing pages.
>> > I don't know how many rows you could get this way, but think it would
>> > approach 13.  It could certainly illustrate the pattern of the numbers.
>> >
>> >
>> > Lloyd Rasmussen, Senior Staff Engineer
>> > National Library Service for the Blind and Physically Handicapped
>> > Library of Congress   202-707-0535
>> > http://www.loc.gov/nls
>> > The preceding opinions are my own, and do not necessarily reflect those of
>> > the Library of Congress, NLS.
>> >
>> >
>> > -----Original Message-----
>> > From: Blindmath [mailto:blindmath-bounces at nfbnet.org] On Behalf Of Joseph
>> > Lee via Blindmath
>> > Sent: Thursday, July 31, 2014 7:45 AM
>> > To: 'Elise Berkley'; 'Blind Math list for those interested in mathematics';
>> > 'Alexa Schriempf'
>> > Subject: Re: [Blindmath] Pascal's Triangle in Braille?
>> >
>> > Hi,
>> > Pascal's Triangle describes a pattern for locating factorials, permutations
>> > and combinations. It is arranged like a pyramid, with the first entry being
>> > 1. Each level of the triangle describes the following: each row represents
>> > each number (1, 2, ... n), and each column describes the permutations. For
>> > example, if someone says, "what is 5 choose 3," then the person would look
>> > at the fourth column of row six (the top row is zero).
>> > I think the simplest way to represent Pascal's Triangle in braille is not to
>> > use the visual layout; rather, start from the leftmost margin, with each
>> > line being a row from the triangle, with columns separated by a space.
>> > Cheers,
>> > Joseph
>> >
>> > -----Original Message-----
>> > From: Blindmath [mailto:blindmath-bounces at nfbnet.org] On Behalf Of Elise
>> > Berkley via Blindmath
>> > Sent: Thursday, July 31, 2014 4:16 AM
>> > To: 'Alexa Schriempf'; 'Blind Math list for those interested in mathematics'
>> > Subject: Re: [Blindmath] Pascal's Triangle in Braille?
>> >
>> > Okay.  I am claiming ignorance here.  I am not familiar with Pascal's
>> > triangle and I cannot get a visual of it.  Can someone help me a little bit
>> > here?  Or, if someone has a simple example in braille, I can pay for a copy
>> > in the mail?  This is interesting.  Elise
>> >
>> > -----Original Message-----
>> > From: Blindmath [mailto:blindmath-bounces at nfbnet.org] On Behalf Of Alexa
>> > Schriempf via Blindmath
>> > Sent: Tuesday, July 29, 2014 7:15 PM
>> > To: blindmath at nfbnet.org
>> > Subject: [Blindmath] Pascal's Triangle in Braille?
>> >
>> > Dear Listers:
>> >
>> > Today I made a Braille version of Pascal's triangle for a student. Needless
>> > to say, because I could not adjust the font size of the Braille, and because
>> > our paper is 11 inches wide, I had some issues with recreating it with
>> > Duxbury and our embosser.
>> >
>> > Pascal's Triangle is a very specific pattern of numbers, and the orientation
>> > of the numbers is carefully placed in a honeycomb pattern. To make 4 digit
>> > numbers fit in the honeycomb slot of the same size that holds a 1 digit
>> > number, typical images of Pascal's triangle simply use a smaller font size.
>> > This does not work when using Braille of course.
>> >
>> > I tried to put this in a table, but the table cells need to be staggered.
>> > In the end, I I simply printed out each row and cut the rows into strips,
>> > taped the strips together where needed, and then glued the whole thing to
>> > cardboard.
>> >
>> > However, while this will work for understanding some of the basics, it won't
>> > help where it's necessary to see how the numbers align physically on the
>> > paper because the inability to change the size and spacing of Braille dots
>> > means that I can't fit a four digit number to fit underneath a 2 digit
>> > number.
>> >
>> > Surely, this learning object must already exist in Braille someplace?
>> > Please advise.
>> >
>> > Thank you so much!
>> > Alexa
>> > --
>> > Alexa Schriempf, Access Tech Consultant
>> > Office for Disability Services
>> > Teaching and Learning with Technology: Accessibility Group Adaptive
>> > Technology Services, University Libraries Penn State
>> > https://sites.psu.edu/aschriempf/ http://equity.psu.edu/ods
>> > http://tlt.its.psu.edu/
>> > http://www.libraries.psu.edu/psul/adaptivetechnologies.html
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> 
> 
> 
> -- 
> Alexa Schriempf, Access Tech Consultant
> Office for Disability Services
> Teaching and Learning with Technology: Accessibility Group
> Adaptive Technology Services, University Libraries
> Penn State
> https://sites.psu.edu/aschriempf/
> http://equity.psu.edu/ods
> http://tlt.its.psu.edu/
> http://www.libraries.psu.edu/psul/adaptivetechnologies.html
> 
>

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