Last Thursday I visited River Valley Waldorf school in Upper Black Eddy, PA. You know that it is not a usual school when you are first greeted by these nice chicken :-)
The school is planning Math Fair in November, so the purpose of my visit was to tell students, teachers, and parents about my crocheted hyperbolic planes and gve some ideas what would work for Math Fair.
Students were impressed with my inaugural Guiness World Record in "hyperbolic crochet" but for now it is not enough time to break the record by the time of their Math Fair. So I suggested to make a large hyperbolic soccer ball using paper templates or to crochet.
To understand hyperbolic plane it is important to understand what is curvature.
When I was in elementary school in fall we had to draw leaves. I always had trouble because I was drawing them like they are flat. But they are not!
Let us carefully cut a thin strip around the edge of the leaf. If I let it freely lay on the table then it is springing up.
If I am trying to flatten the edge s that it would all be on the table, then I notice that the edge is "too long" - it does not come together at the point but is starting to overlap.
Try this experiment with other leaves! You will find that there will always be some overlap. It is because leaves are not flat but they are curved.
Let us look at a crochet pattern. It is called African Flower (pattern/tutorial). It lays nicely flat on the table because it is forming a hexagon. If I make in this flower one petal less, I have a pentagon which does not lay flat on a table - it curves up or down.
You can imagine how by adding more these five petal flowers I would get something very much like a ball. If you do not believe me - make 12 such motifs and then join them together at the edges. This will make your crocheted dodecahedron. Your surface will be closed like a ball or sphere. This type of curvature is called positive curvature. Since sphere is the same everywhere - you can see it has constant positive curvature. Stitching together pentagons and hexagons will end in a shape of traditional soccer ball. Now the pattern has chnaged but I hope you still remember this:
Now let us use the same African flower pattern but this time instead of six petals let us make seven.
This flower also does not lay flat on a table but is curved differently than the flower with five petals - there are some "ups and downs". It is more like curly parsley or cale. This type of curvature is called negative curvature. If this negative is the same everywhere (or we say - constant) then we have hyperbolic plane. In the sense of curvature we can say that hyperbolic plane is something opposite to sphere. You can try to make your own hyperbolic plane using template of "hyperbolic soccer ball". (here is another version , you can see the animation how it will grow here.)
This time I crocheted one - middle motif is with seven petals but I used only one color yarn to stress that there are seven hexagons around the heptagon.
And this is my crocheted version of "hyperbolic soccer ball".
Another crochet version could be made using just seven petal flowers alone - this way curvature will show off faster.
In class students made their own hyperbolic soccer balls from paper templates. We did not have time to join them all together but hopefully that will be done for Math Fair.
|photo Derek Smith|
|photo Derek Smith|
Waldorf school is ver hands-on oriented and of course children wanted to find a practical applications of hyperbolic planes. Some said they will be making hats in shape of pseudosphere, some will make scarves. I am very much looking forward to see what ideas of our meeting will be used in Math Fair.
|photo Derek Smith|
My thanks to Derek Smith for organizing this trip and to all my very nice audience in River Valley Waldorf school.
If you would like to see some more ideas - Crocheting Adventures with the Hyperbolic Planes are now available in Kindle version. I have not seen it myself and not very happy about not knowing that the book will be made in Kindle version - it was not written as e-book, so I wish I would be given an opportunity to edit it accordingly.
On my way back to Ithaca I stopped at Delaware Water Gap where I took this picture.