Tag: mathematics

Tau-Day (yesterday)

Tau-Da!

I completely missed the celebration of tau yesterday as well as being completely ignorant of a movement among mathematicians to replace my favorite constant, pi.  I have spoken previously (once) here about pi and even use the first few digits of it as part of my username here at WordPress.   At least once a year, on Pi Day, which corresponds to March 14th (or as close to 3.14 as you can get on our calendars), I celebrate the never-repeating, endlessly enlightening expression of the ratio between a circle’s circumference and it’s diameter … or wait, is that it’s radius.

I received a Tweet that intrigued me, entitled ‘Mathematicians Want to Say Goodbye to Pi‘ with an accompanying link.  I read the article, but what really held my attention was an inserted YouTube video from someone named Kevin Houston (with a British, not Texan, accent).  If you enjoy math, take a few minutes to watch his video.

So, since Pi Day is celebrated on March 14th, and 2π is roughly 6.28, it follows that celebrating tau should occur on June 28th.   Or so the tau enthusiasts hope.

I’m still on the fence, preferring pi for the moment; although, I agree the use of tau has its merits in simplicity and beauty.

Constant Celebration

Can you guess what my favorite mathematical constant might be?  There is a clue in the URL address of my blog.  Still unsure?

This mathematical constant whose value is the ratio of any circle’s circumference to its diameter.  It is an irrational number, which means that its value cannot be expressed exactly as a fraction m/n, where m and n are integers. Consequently, its decimal representation never ends or repeats.  It is also a transcendental number, which implies, among other things, that no finite sequence of algebraic operations on integers (powers, roots, sums, etc.) can be equal to its value.

Yes, together with other math-loving geeks out there in the universe, I’m celebrating Pi Day.  And if WordPress behaves itself and publishes this as I’ve scheduled it to, at exactly 1:59 pm (Central time), I will have succeeded in my mathematically constant celebration.

Pi pie
Pi pie

Algorithmic Crochet

I forgot in my last post on my crocheting technique, Back to Math Basics, to explain further how I use the following table and how I arrived at it’s contents:

Row x2 x3 x4 x5 x6
16 32 48 64 80 96
17 34 51 68 85 102
18 36 54 72 90 108
19 38 57 76 95 114
20 40 60 80 100 120
21 42 63 84 105 126
22 44 66 88 110 132

I used some algebra (or possibly finite mathematics) to achieve the process for inserting the six additional stitches equadistant around the circumferences of the cap.

Let the row be represented by a; let the incremental stitches be represented by b; and, let the insertion position for the additional stitches be represented by P.  Then,

P = ab-1

Using the table above, at the beginning of a round, I stitch a-1 stitches and insert the a stitch in the same stitch.  For row 16, I started off with 15 stitches and put the 16th stitch in the same stitch as the 15th, then I continued counting 15 more stitches (or to the number 31, since that is ab-1 or (16×2)-1, and inserted the second additional stitch, which is also the 32nd stitch, in the same stitch as the 31st stitch.  How many times can I type stitch in a sentence?  🙂

When I reach my safety pin which marks the end (or beginning) of the round, I should have reached the 96th stitch, after which I slip stitch into the first stitch of that round.

Algebra doesn’t always lends itself seamlessly to application in crochet, but an algorithm works perfectly.  For example (using no particular programming language, but rather just generic easily understood syntax):

Row = 16
MaxRow = 22
IncreaseBy = 6
Stitch=1
AddStitch=1

While Row < = MaxRow
While AddStitch <= IncreaseBy
While Stitch < Row
Stitch=Stitch+1
EndWhile
AddStitch=AddStitch+1
EndWhile
Stitch=1
AddStitch=1
Row=Row+1
EndWhile

I think that algorithm works, however clumsy it may appear.  I’m sure I could do it with less loops and/or recursively, but I’m too far removed from my programming days to dredge up those memories.  I may research a bit to remind myself of some more aesthetic algorithmic techniques and revisit this in a later posting.

Still, I find it fascinating to confuse what otherwise could be a boring bit of crocheting.  Besides, I always love to tout all the math my fellow students complained about in high school, whining that they would never use algebra or geometry or trigonometry, etc. in the ‘real’ world.

Ha!

Back to Math Basics

Sunday afternoon became more distracting as it approached evening.  Aside from the numbness and tingling which reasserts itself every few minutes, I find it difficult to count stitches and determine multiples of double digit numbers in my head while remembering the end goal of max stitches for that row all while the rest of the family watches a movie or taunts the Rotts into playing boisterously.

At row fifteen, I stopped and took a break for a bit.  I read a few pages in Grand Conspiracy.  I then found a piece of paper to write out the next seven rows numerical stitch pattern.  For example, until row twenty-two, I need to increase each row by adding six stitches evenly spaced around the round.  I wrote the following quick chart to aid in my stitch counting:

Row x2 x3 x4 x5 x6
16 32 48 64 80 96
17 34 51 68 85 102
18 36 54 72 90 108
19 38 57 76 95 114
20 40 60 80 100 120
21 42 63 84 105 126
22 44 66 88 110 132

I made it to row eighteen last night before retiring to bed.

Eighteen Rows or 108 Stitches Around
Eighteen Rows or 108 Stitches Around

Part of the reason I enjoy crocheting, or music (which is tangential I know) has to do with all the finite math involved with the patterns.  And the best part of all, at least when working a circular crochet pattern is the chance to use my favorite mathematical constant.  Stretch your memory back to the days of algebra and geometry and remember the simple formula for determining the circumference of a circle:

Diameter of 10.25 inches
Diameter of 10.25 inches

Can’t remember?  Well, let me remind you using the photo above.  If the diameter of a circle is 10.25 inches, the circumference is the diameter multiplied by the constant pi:

C = dπ

Or, as illustrated above:

C = 10.25 * 3.14159

C =  32.2

I have four more rows of increasing before I crochet a band of a half dozen single crochets (with no increases in stitches).  After that, I start decreasing.  The pattern reduces to a head band circumference of 18.25 inches, which is too small for Rachelle’s inflated ego, er I mean head.  Her cranium has a circumference of over 22 inches.  So I’ll have to do yet more math to determine the proper stopping point during the reduction.

I’ve decided not to take this Brimmed Cap project with me to work today, even though with the vanpool I have over an hour I could be crocheting to and from work.  Mondays (and Fridays) I usually have to tote quite a few things with me (like a week’s worth of lunches and a laptop).  If I don’t finish the cap this evening, I’ll probably take it with me on the commute Tuesday.