Wednesday, July 31, 2013

Experimenting with Slide Togethers

After creating the slide togethers by George Hart, I decided that I wanted to create my own models.

Here is the Product of My Experimentation
 

I used cardstock for all of the models.

I created a flower blossom slide together with 60 degree angles.  The model was hard to put together because the edges did not line up correctly...maybe it was an angle problem but the interior of the design looks like an equilateral triangle. Oh well, it's a pretty flower and I am posting it for anyone who is interested in a challenge.  But be forewarned, it will collapse while putting together...use glue or paper clips and hold tightly by cupping the model in your hand when constructing the model.  Here is the .Studio file.  Cut 20 shapes for one model or two pages with extra pieces.
https://docs.google.com/file/d/0B7oGIyVDbRGYOGxUZFVLel9NUE0/edit?usp=sharing

Flower Blossom Slide Together

Butterfly slide together with 72 degree angles...notice the pentagon shape inside the flower petals.  I like this model the best. It will collapse a little when putting together but the effort is well worth your time and patience.  Here is the .Studio file for cutting the model. Cut one page (12 shapes).
https ://docs.google.com/file/d/0B7oGIyVDbRGYdDJ0UndTcTh2d0E/edit?usp=sharing

Butterfly Slide Together

Star flower slide together with 60 degree angles. It is a nice slide together.  It is a little difficult to make but it is satisfying when completed.  Here is the .Studio file. Cut two pages (need 20 shapes) https://docs.google.com/file/d/0B7oGIyVDbRGYSU1KR2ZiUGtJb2c/edit?usp=sharing

Star Flower Slide Together

Star slide together. This model is easy to put together...great for students...no frustration factor.  The model has 72 degree angles. I actually made this model twice. Here is the .Studio file. Cut one page (12 shapes).

Here is the PDF.
https://drive.google.com/file/d/0B7oGIyVDbRGYSUpadndKaTZFWkk/edit?usp=sharing

Here is the .Studio file.
https://docs.google.com/file/d/0B7oGIyVDbRGYOGhfWTR6alpVUk0/edit?usp=sharing
 First, all one color.

Star Slide Together

 Next with patriotic colors.  You can not color this model with three different colors not touching as with the origami Buckyball which I explained in a different post..  Try it and see.
 
Patriotic Slide Together

#silhouette cameo #slide together #3D models


Sunday, July 28, 2013

Slide togethers by George Hart

George Hart is my hero.  His website http://www.georgehart.com/ is fabulous.  His objective is to make math cool and I think he achieves this goal.


I have designed the cutting files for all seven slide togethers. Please me mindful of George Hart's work. I copied the work to promote his mission of making math cool and help others who want to explore and enjoy his work.  As he says on his website..."the templates below may be freely copied for educational purposes. Creative teachers can undoubtedly incorporate these constructions into classes of different levels in ways which I would never think of" http://www.georgehart.com/slide-togethers/slide-togethers.html

The photographs below have a tape measure in each picture so that you can see the size of each model...just in case you would like to resize. Also, some of the templates maximize the amount of paper if you are cutting this out for a class and have extra pieces. Use cardstock for making the models.

This is the easiest model to make but it tends to collapse and fall apart when put together. (Use paper clips when putting together to alleviate this problem.)
Here is the Triangle slide together .Studio file - 20 triangles make one model.  Cut two pages (There are16 triangles on one 8 1/2 x 11 page.so be aware there will be a lot of extra pieces.)
https://docs.google.com/file/d/0B7oGIyVDbRGYZGx0cnAxUVpwbzg/edit?usp=sharing



Here is the square slide together.  Need 30 squares. Cut 3 pages. (There are 12 squares on each page so there will be extra pieces.) This is the one I did with my pre-algebra class.  A nice satisfying model to make.
https://docs.google.com/file/d/0B7oGIyVDbRGYVWpwb2dUbzlOam8/edit?usp=sharing



This pentagon model takes awhile to put together. The paper has to be bent in order for this model to fit together. Here is the pentagon .Studio file. 12 Pentagons are needed.  Cut two pages.
https://docs.google.com/file/d/0B7oGIyVDbRGYMG9DelJzd281RTQ/edit?usp=sharing



I would recommend this hexagon model as a class project.  It slides together nicely and it holds its shape...also no bending of paper is required. 20 Hexagons are needed.  (There will be two extra pieces.) Cut two pages. Here is the .Studio file for the hexagon
https://docs.google.com/file/d/0B7oGIyVDbRGYdTJHa3NONzJQcDg/edit?usp=sharing



This decagon model requires a lot patience and three dimensional awareness. Each vertex requires a few steps...the slits are bent flat to slide together and then peeled away for the inner slit to fit together.  Cut 2 pages. (There will be extra pieces.) Here is .Studio file https://docs.google.com/file/d/0B7oGIyVDbRGYd2p3QmxrdFA0ekk/edit?usp=sharing



This star slide together is difficult to make.  It requires a lot of patience and three dimensional awareness. The slits are small and can tear easily so be gentle. The big slit goes into the little slit and then the paper is folded back to allow for the slits to align. Again, this requires bending of the paper. Cut 2 pages for one model. Here is the .Studio file
https://docs.google.com/file/d/0B7oGIyVDbRGYWlR5bThGc3Y2QjQ/edit?usp=sharing



This decagram slide together is the hardest model to make.  I could not make this model in one day because it is very frustrating but it is really beautiful when it is completed. Cut 2 pages for one model.  Here is the decagram .Studio file.
https://docs.google.com/file/d/0B7oGIyVDbRGYSVpYZ1o3dWdleWc/edit?usp=sharing



Please comment if there is a problem with the .Studio files.  Thank you!

#Slide Together #Cameo Silhouette



Friday, July 26, 2013

Origami Buckyball Puzzle

Challenge: Make a Origami Buckyball ( A dodecahedron - 20 vertices, 30 edges, 12 faces (regular pentagons)) using three colors so that no two units of the same color touch.

I love puzzles and this challenge took me a few hours to solve. In order to solve this puzzle, I had to do a little exploration of the materials.  Given three different colors that meet at a point...how many combinations can there be.  As you can see from the photo below, only two.



The next problem was combining these pieces to create a pentagon.  How many different combinations are there?  Three combinations.



The final problem is how to put these pentagons together to create the dodecahedron, where all vertices are different colors and don't touch one another. After trying a few combinations, I discovered that I needed more theory.  It is a circuit problem.  A Hamilton Circuit in particular.



Starting at a point in the middle.  You need to draw a line so that you do not lift your pencil.  This creates a circuit. 


Next you need to alternate the colors of the line. Notice that each vertex has three lines emanating from it and that there are three different colors.  Yes...problem solved!

The final result.


I cut out the 30 squares using my Cameo Silhouette.  I am not including the cutting file because it is so easy to make a 3 inch square and fill a page.  Here is the origami directions for making this Buckyball http://www.nisenet.org/sites/default/files/catalog/uploads/9728/buckyballorigamiv02.pdf

#Hamiltonian circuit #Origami puzzle #Cameo Silhouette

The Coolest sliceform - Hyperbolic Paraboloid

The Museum of Mathematics in NYC created this hyperbolic paraboloid sliceform.  It is amazing to see the form take place as you put it together. It lies flat after you put it together! You have to love it!

My daughter loved the fact that when the parabola lies flat it shows the integrals for the area under the curve...a good sliceform to show to a calculus class.  For the less mathematical people,  the area under the curve can be explained as...what is the area of this three dimensional parabola?  The area of multiple two dimensional rectangles added together will result in the approximation of the total area of this 3 D parabola.



Here is the link to the pdf of this sliceform. http://momath.org/wp-content/uploads/Hyperbolic_Parabola_Model_85by11.pdf

If you have a Cameo Silhouette, you are in luck because I created a .Studio file for this sliceform so you do not have to do any of the cutting by hand.

Here are the two .Studio files.  Make one copy of each. 

https://docs.google.com/file/d/0B7oGIyVDbRGYNlhpVWs4S09Lelk/edit?usp=sharing

https://docs.google.com/file/d/0B7oGIyVDbRGYVndRZnJPbl8yRGs/edit?usp=sharing

After cutting the file, there might be some confusion as to how to put the pieces together.  I hope this explanation helps...otherwise...try to label the pieces according to the pdf file. The pieces slide together in the following order.  Pair up the pieces of file #1. Do the same for File #2.  There should be one piece that does not pair up - a rectangle. Take the two rectangles and slide them together in the middle.  All the other pieces are now trapezoid shapes. There should be six of them for File #1. Slide the pieces together from the outside in starting with the smallest length trapezoid using the middle slit. The highest points will be on the opposite sides. Repeat with pieces of File #1.  Do the same for File #2 pieces. The opposite sides will have the tallest side on the outside.







Thursday, July 11, 2013

Tangrams



This post might be boring if you don't have a Cameo Silhouette but I am posting it anyway for any math teachers out there that need the cutting file for tangrams.  Tangrams are an excellent activity to give students who have completed their assignments early and are looking for something interesting to do. Here is an excellent PDF worksheet with answers.  The tangram area problem is a nice challenge for geometry students.

http://math.schaubroeck.net/files/tangram_ANSWER_KEY.pdf

Here is my .Studio file to cut out the tangrams for students. Yes, I know the students can cut them out but as a time saver...you can cut them out ahead of time.

https://docs.google.com/file/d/0B7oGIyVDbRGYU05YeXFZbWZlMjg/edit?usp=sharing

#tangram #Silhouette

Tuesday, July 9, 2013

Origami Octahedron and Gyroscope

I am working my way through the book entitled "Beginner's Book of Modular Origami Polyhedra: The Platonic Solids".  The book itself has a lot of interesting origami models.  However, the directions are terrible.  The book misnumbers steps, the diagrams do not show which side of the paper to use and it doesn't tell you how much origami paper you need to complete the model.

This octahedron was simple to make with six water bomb bases. The pieces slid together in about 20 minutes.

I also made this origami gyroscope.  I think it turned out too big.  I used a 4x4 square. Next time, I will be more mindful of the size. Another gripe about this book...they should suggest paper size.  I do like the square cut out in the middle thoug...a window to the other side.


Monday, July 8, 2013

Sliceforms are my new obsession




I love sliceforms.  What are they you say?  Well, they are exactly what the term says...slices of a form.  For example, I decided to make a sliceform of a sphere.  I used my designing software with the Cameo Silhouette and created this sphere. I have included the PDF and .Studio files if you would like to make sliceforms at the end of this post. They would look great as a decoration for a party or as an ornament. The shapes took me hours to make so please pass them along and appreciate what math can do!
 Marley loves it too! Amazing that it lies flat when put together.
 
Sliceform Sphere


I saw a sliceform of a torus on the internet.  I made it and then decided it was not pretty enough so I created my own version. The pink torus is the scalloped version and the blue torus is the zigzag version. This time the toruses makes me think of flowers instead of food (a donut).  I love the way the variations of color are created by the shadows of the slices and the spiral in the center.  There is so much going on mathematically as well as artistically.  It makes me so happy!




For the mathematical people there are villarceau circles which create this shape.  Here is an explanation from Wolfram Alpha.

Villarceau Circles

DOWNLOAD Mathematica Notebook
TorusCirclesVillarceauCircleXSections
Four circles may be drawn through an arbitrary point P on a torus. The first two circles are obvious: one is in the plane of the torus and the second perpendicular to it. The third and fourth circles (which are inclined with respect to the torus) are much more unexpected and are known as the Villarceau circles.

The cube sliceform makes me think of the dividers in a box for glassware so that the glasses do not touch one another.  They have practical applications. Sliceforms are amazing!

Sliceform Cube
 
Here is the PDF of the sliceforms.
 
Here is the .Studio file.  I used cardstock to create my models.




#sliceforms #Silhouettecameo #Villarceau #torus