Saturday, December 6, 2008

Basic Geometry II

Welcome back to math class. :D
Actually, no.

In addendum to the Easy Softie Ball in 12 Pieces, there exists an even easier way to make a ball in six pieces. The 6-piece version, like the 12-piece, only requires the copying and cutting out of one form, but there are only half as many copies to make. Ergo, easier. Yay!

Now, if you're interested in the math behind the form, a 6-piece from peels is actually a little more complex than a 12-piece from pentagons. The 12-piece ball isn't actually a ball in Euclidian geometry: it's a dodecahedron that becomes a ball when you sew it up and stuff it because the pressure of the stuffing inside applies evenly across the planes, seams, and vertices. The 6-piece ball, on the other hand, is an actual, true sphere in design and execution. This is because it employs the relationship between the circumference and diameter of the circle formed by the cross section of the final sphere at maximum diameter.

And you thought this was a post about softies. >w<

By now, you might be asking "hey Ku, what am I supposed to do with all these plush balls?"

Inexpensive, unbreakable, toddler-safe Christmas ornaments. Ta-da~!

Marian said...

Iḿ so happy I found your blog. Immediately added to my bloglines.
I'm new in the world of sewing. I recently got a sewing machine as a gift and for the first time in my life I sew something...quite rudimentary my firrst item. But now a whole new wolrd has oppened up. Im moslty a ceramist and handfelter.
I've been wanting to make an apple...yes, and apple. Why? dont know. I just want to make a softie apple...and can't figure out how (ok, Im not dumb...just a bit maybe, remember...new world. Im a psychologist, this is new territory!).
So, you have tutorials to "fruits" and other things so this might give me an idea.
THANKS FOR SHARING!!
Will check back often when you update!

Florcita

Ku said...

Hello and welcome! It's nice to meetcha. XD Thanks for stopping by!

Sewing is great! I'm no pro, either, but I've made a few Halloween costumes and such in my life. I actually don't use a machine very often since I work in felt, which isn't the most machine-friendly fabric... If you decide to make anything from this blog, I would suggest hand-sewing with a running stitch , or using cotton fabric instead of felt. ^^;

Handfelting~ *drool* @,@ I've always wanted to try that since I first read the Sinco books, but there's no source of wool around here. >.< Maybe I'll break down someday and order a needle and fibre online. :)

It's funny that you should mention an apple... I've been trying to figure that one out for a while now. ^^; Softies have the highly limiting characteristic of being entirely convex. That is, when you put the stuffing in, everything wants to push out, so getting "dimples" such as those in the top and bottom of an apple is actually a big conundrum for me. (Haven't given up yet, though... I'll let you know if I ever get it right!)

Doumo sa! Thanks again for popping by! XD

Nikki said...

Yanno, an easy-peasy-probably-cheating way to get a dimple would be to make a knot at once end, run the thread through the entire dealy, and pull it taut through the other end, then knot again.

Actually now that I think about it that would just make a blood-cell-shaped blob...unless you could somehow vary the concentration / density of stuffing in the dimpled and non-dimpled ends. Hmmmm...gorrammit, you've gotten me all intrigued now.

BTdubs, I'm a lazy hobo and thus am waiting 'til I get home fer Yule to send you my anteater pattern. Also the scanner is at home, so... ;(>.>)

ミJean★Claudeミ said...

ある日、一つのいわゆるポストにすがたをあらわす物gat

Ku said...

HAHA, I intrigued you! Nya-na-na~

You're right... it does make a blood-cell blob. But, hey! Now we can make softie blood cells! XD The "variations of stuffing density" idea is interesting. I'm thinking, what if you sewed the shape with two, three, or four upright "chambers" together, such that the stuffing would be forced to apply pressure along the seam boundaries formed by the panels on the inside? ....if that makes any sense? o.o

PS: BOX AT YOU. It's not time-sensitive, so no worry if you donnae get it before your Yule pilgrimage.

Ku said...

ジェアンクロード様、ヨーホ！いったいなんなんだろう？O.o

ミJean★Claudeミ said...

I Purr-Furr to Craft said...

looks great as Christmas tree ornaments

Katherine said...

I think this is a beautiful geometrical picture. I was going to ask you how you figured out what the radius (R) for the circle the pattern arc is taken from, I worked it out though, and I came up with R=(l*l + w*w)/(4w), which seems to work out when I lay out the circles on graph paper. Thanks for inspiring some morning math.
-Katherine from OneInchWorld.com

jessimonster said...

Thank you, thank you, thank you!!
Your blog has everything I've been looking for today, and more! My home made Christmas will be perfect this year! Thanks again!

Ku said...

Katherine and Jessi, nice to meet you! :D Thanks kindly!

Ellie said...

this is PERFECT! you know what im going to do with this pattern? im going to make a 'boo' doll. remember those ghost from mario? XD
i had no idea where to start since idk how to make a ball, so many, many, thanks!
btw. i have to say, i kinda started zoning out when you talked science/geometry or whatever that was. lol. that kind of stuff just goes over my head! XD

judithchenartworks said...

aww this is very useful..thx :)

awesome layout btw, how did you make one? xD

http://judithchenartworks.wordpress.com/

xoxo

Judith

SARVARAKSHANI said...

i was googling for softie tutorials and found your blog. you are very good and i love ur dieas. thank u very much for posting them.

Lee said...

Thank you so much!! I stumbled across your instructions just in time for making a huge disco ball prop out of material for an upcoming show!
Thanks so much for the easy instructions.

So grateful to you! I was able to volunteer today with 20+ at-risk kids from http://www.anewdaycambodia.org/ showing them how to make a Christmas ornament http://picasaweb.google.com/canadasue/2010ChristmasAtANewDayCambodia?authkey=Gv1sRgCOPA6uK9vqn4Ow&feat=directlink

because it was in cotton not felt it threw off the "math" a bit but the boys were delighted with the "Christmas Football" tee hee!
Thank you x 1,000,000 You made some kids really happy today!

MayuChan ~まゆちゃん~ said...

I was looking for a tutorial like this for so long. I'm so happy I found this, thank you soooo much! ♥

Ku said...

Glad to be of service. :3

Martha Murphy said...

Yay! I'm going to elongate it to make a watermelon. I thought I was going to have to draft it myself!

Martha

Ku said...

Wow, people are still finding this site? I feel special. c: Thanks, and hope you get a nice watermelon out of your work!

Joshuacliftojm said...

Hey, I've been working on writing a document giving mathematical definitions of a variety fabric ball designs (specifically for juggling) and I was very glad to find this blog post as it was the first (and so far only) confirmation I have found for my theory for the design of the beach ball (as I call the type of design you're discussing). Commenter Katherine's formula for the radius of the curve of the beach ball panel (or "peel") also helped a lot because I had to use trial-and-error before. Thanks, Katherine!

I'm interested in learning where your theory for the shape of the peel comes from. I arrived at the theory empirically and I don't know enough math to prove it. I found an alternate theory recently that does not use a circular curve, but one based on calculating the circumference of a sphere at different latitudes. So far I have found very little information on making beach balls (technically hosohedrons) out of flat panels, and you are the only one who has said to use a circular curve. Some patterns I have seen use what appear to be other curves, or hand-drawn curves, and some even use different width:length ratios. Are you interested in discussing this?

By the way, here is the title image for my document that shows SketchUp-generated images of all the designs I have defined (http://www.joshuaclifton.com/CGBalls.jpg), and here are photos of my proof-of-concept balls (http://www.joshuaclifton.com/Samples.jpg). I used the beach ball concept to apply curves to triangles and squares for the spherical tetrahedron, cube, and octahedron. I still don't like that I can't be sure if it is an optimal solution, though.

Thanks for the informative blog post!

Ku said...

Heck yeah, I'm interested in discussing this, but damned if I have the time... I'm really sorry, but this is my final semester of a grad program, and I'm up to my ears in other responsibilities. I'd really love to see your master thesis, though. :3

The circular curve just kinda fits my understanding of a three dimensional object... because the seams travel the circumference of the final shape, it made sense to me to make it a portion of a circle rather than a parabolic curve of some sort. If there's math to support that choice, I don't know it, either. ;_;

Joshuacliftojm said...

Thanks for the reply! I wish you well in your grad program. I had the same reason for using a circular curve. I just makes sense as I am trying to approximate a sphere. It's also simple.

Here is a PDF export of the current draft of my instructional document (http://www.joshuaclifton.com/How%20to%20Make%20Juggling%20Beanbags.pdf). It's 12.4MB due to being 159 heavily illustrated pages. I may not leave it on my server for long as I may someday publish it and I don't want it to travel too widely.

By the way, good job on the photography and illustrations in your blog posts. I've only read the two related to making fabric balls, but you take attractive and effective photos of your projects and and I like the design of the text on the illustrations.

vaneea said...

Hello,

I am trying to make a giant ball.
I am thinking about 50-60cm in diameter.

Which pattern would you recommend me to use? This one or the 12 piece one?
If you recommend this one, how many pieces do you think I should use?

thanks so much!

-Vania

Ku said...

Hi Vania!

Boy, that's a good question. If you use the 12-piece pattern, it'll be more stitching, but the pieces will be smaller and all the sides will be straight, so setting up the pattern on the fabric will be simpler.

If you use this one (6-piece), you'll have fewer pieces to have to deal with, but each one of them will be bigger, and they'll have a very specific curved sides that you'll need to get right. Regarding how many pieces to use for a large ball if trying this pattern, the answer is always going to be 6. You don't have to add more pieces for a bigger ball, just scale them up in size. (Same rule for the 12-piece.)

Personally, I'd go for the 12-piece version. The more colors I can jam into a project the better, so the chance to use 12 different fabrics trumps 6. But that's just me. :3

Joshuacliftojm said...

Vania,

If you use the 12-piece design (the dodecahedron), here are the formulas for the dimensions of the pentagonal panels:

Pentagon Side Length = ball diameter × π ÷ 8.1554
Pentagon Radius = ball diameter × π ÷ 9.5872

Of course, the fabric may stretch a bit. When I made mine with denim, the ball ended up being 2.5% larger than the calculation predicted. If you're picky, you can compensate for that by dividing the result of the formula by 1.025.

If you want to make the 6-panel design (a hosohedron/beach ball), the dimensions of the peels are as follows:

Peel Width = ball diameter × π ÷ 6
Peel Length = ball diameter × π ÷ 2
Curve Radius = (W^2 + L^2) ÷ 4W
Distance between compass points = 2r – W

Draw a line equal to the result of the fourth formula and use its two endpoints to position a compass. Adjust the compass to the necessary curve radius and use it to draw the curves from those two points. Or use a computer application such as SketchUp (which is free) and use the endpoints as the centers of two circles. To ensure you drew the peel correctly, measure it and make sure its length is half your target circumference (which is diameter × pi) and its width is equal to the circumference divided by the number of panels you're using.

If you want to make a beach ball with more than 6 panels, just substitute the number of panels for the "6" in the first formula. The other three formulas will still work as they are.

---Joshua

Ku said...

Thanks, Joshua! (I'm still keen to see your full thesis -- sounds like you knocked the math out of the park!)

Anandita Sathyanarayana said...

Is the dashed line where we should sew?

Samppa Tolvanen said...

Oooh,

Thanks Shishi and Joshua, needed the plans and math because it's n+1 years since my latest geometry classes. ;)

Let's see if it's a friends dog that gets frustrated this time, not them fixing her toys: Denim fabric and heavy duty polyester sawing thread should keep her busy a long time :D

Remember to leave enough space for the strengthening stitches and the thing is easiest to put together from 3 piece blocks rotated 180 relative to each other. My sawing machine barely survived.

Thanks again!