Monday, June 27, 2022

A STEM Project: Three Cube Sliceforms Imbedded with a Diamond, a Sphere, and a Sphere/Diamond Combo

Three Cube Sliceforms Imbedded with a Diamond, a Sphere, and a Sphere/Diamond Sliceform Combo

Sliceforms are visually pleasing structures and paper is one of the only ways to create them and that is why I love them so much. Paper needs to be carefully bent to slide each slice into place and rigid materials such as wood or a hard plastic will not suffice.  The resulting figure is a design which can lay flat.
In this blog posting, I have created three different cube sliceforms.  In order to make each sliceform, start with sliding the two center slices together.  Proceed with sliding the imbedded sliceform together. Finish with the outer cube slices.

A Cube Sliceform Imbedded with a Diamond Sliceform

A Cube Sliceform Imbedded with a Sphere Sliceform

A Cube Sliceform Imbedded with a Sphere Sliceform Imbedded with a Diamond Sliceform

Wednesday, June 15, 2022

A STEM Project: Making a Torus Sliceform

Two versions of a Torus Sliceform
On the left is a twenty four slice version and on the right is a sixteen slice version

The torus is a beautiful design. I am fascinated by the dynamics of its construction. Toruses are made with two types of Villarceau circles which are half moon slices.  The two types of Villarceau circles have opposite side slits that are slid into one another to form the torus. Toruses must be made with a flexible material because the Villarceau circles needs to be bent and manipulated to be slid into one another.  In the past, I have altered the edges and angles of Villarceau circles and I created two beautiful flowers, a chrysanthemum and an amaryllis flower

Recently, I discovered this paper, Building a Torus with Villarceau Sections by Marıa Garcıa Monera and Juan Monteabout from the University of Valencia, published in the Journal for Geometry and Graphics Volume 15 (2011), No. 1, 93–99.  and I wanted to perfect the toruses that I made in the past. The paper explains the mathematics behind the creation of the torus. Using their formulas, I was able to create an accurate sixteen slice version of the torus. I have included the angle measurements in my files below.  In the past, I felt that my sixteen slice torus looked good but I felt that the angles in the Villarceau circle were not accurate. I didn't know the formula to correct the inaccuracy. After discovering this paper, my intuitions were correct. I discovered using their calculations that two of the angles of the sixteen slice torus were off by a tenth of a degree. This goes to show that a slight difference in a complex system can make a big difference. 

I have included two versions of the toruses in my files.  The twenty four slice version is a more complex design because of the number of Villarceau circles. The sixteen slice version is easier to put together and I recommend completing this design before attempting the twenty four slice version. The cutting and assembly of the sixteen slice version will take about thirty minutes to complete.  The twenty four slice will take approximately an hour to put together. Please note, it will require a lot of patience when the final slices are stretched and maneuvered to be put into place. Here is a basic tutorial of the weaving of the Villarceau circles, Once the stack of slices are woven together, the torus must be made into a donut shape and the corresponding slices need to be slid into one another.  It is difficult to photograph and I recommend googling a YouTube video on how to make a torus sliceform for further instruction.

Here is the PDF.  I used 65 lb cardstock in two different colors, one for each type of Villarceau circle. I recommend double cutting the pattern as the slits do not cut precisely. It was frustrating and time consuming for me to cut the hanging chads after the Villarceau slices had been cut with the Silhouette after just one pass.

Here is the .Studio file.

Here is the SVG.

Saturday, June 4, 2022

A STEM Project: Coding Waclaw Szpakowski's Rhythmical Lines in TurtleStitch

Waclaw Szpakowski B13, 1926, Museum of Art in Lodz
Coded with TurtleStitch and embroidered with a Brother PE800 Embroidery machine.

Waclaw Szpakowski B6, Series B, 1924
Coded with TurtleStitch and embroidered with a Brother PE800 Embroidery machine.

Waclaw Szpakowski (1883-1973) was a Polish architect and engineer who designed abstract drawings with ink and grid paper.  He began his drawings at the age of 17 and refined them over his lifetime. His ink drawings have very precise properties with straight lines drawn at right angles to one another which never intersect. Szpakowski created his drawings with lines that were 1mm thick and 4 mm apart. The ink drawings take on a rhythmical pattern starting on the left side of the page and ending on the right side. The movement of each design creates a maze of lines. The viewer of his drawings will ponder how the drawing was created by following the lines with their eyes to see where the line will take them. Where is the repeat in the pattern? Is the pattern flipped or was it repositioned? The patterns have a lot of movement in them and as a result, optical illusions of different patterns can be seen.

Waclaw Szpakowski published an album of sketches when he was 85 years old called "Rhythmical Lines"which displays the simplicity of the line and the geometric relationships between the lines. He described his works as "drawings of linear ideas" and by viewing his works, it shows the visual harmony of the "mathematical order of the universe" which includes symmetry and rhythmical balance. 

I coded a few of Waclaw's Szpakowski's designs in TurtleStitch and decided that I could create a basic TurtleStitch template to code Szpakowski's drawings because I used the same commands repeatedly. Here is the template that I created. 

The basic template that I created includes the following:

+ A size(scaling) factor and a width variable so that the design can be resized easily for embroidery with different hoop sizes. 

+Variables with basic values. In my early attempts to code Waclaw's designs, I used mathematical equations to move the turtle.  Subsequently, I determined that it is easier to have a set value for the turtle to move because of debugging. It is easier to see where the incorrect value is without having to do a computation.

+ Many of Waclaw's patterns are repeated from left to right across the design and a block command can be written for the code. By pointing the turtle in the upward position (point in direction 0), the block can then be repeated or flipped easily.

+ I created two simple turn and move blocks for the turtle, a "Turn Left 90 Degrees"  block and a "Turn Right 90 Degrees" block. Both of these blocks turn and move the turtle in their corresponding direction and allow for resizing and flipping. I used a flip factor equation so that the design can be flipped and the TurtleStitch code can be reused by assigning a flip value. 

Before I explain my flip factor equation, I will explain how an object is flipped over the y-axis. 

When an object is flipped over the y-axis from Quadrant I to Quadrant II, the x value changes from a positive value to a negative value.  The y value remains the same. 

This fact holds true when an object is flipped in code. I created an equation that will flip a point from a positive location to its negative location. Since all of the turtle movements are at right angles or 90 degrees to one another in Waclaw's designs.  A simple equation can be made to turn the turtle.

Turn Right 90 Degrees

I created the above "Turn Right 90 Degrees" block to turn the turtle to the right. There are two variables, "flip1" and "flip2" which determine the location of the turtle.  (Please note that the steps and size factor variables in the equation determine the distance and scaling factor that the turtle will move.)

Unflipped Position 

The "flip1" variable must be a 1 and the "flip2" variable must be a 0 for the turtle to turn to its "unflipped" position of 90 degrees.

The "flip1" variable must be 1 for the turtle to move to its "unflipped" position.

Flipped Position

When flipping the turtle, it must move 180 degrees to achieve the flip plus the original 90 degrees for the turn.  The "flip1" variable must be -1 and the "flip2" variable must be 360 for the turtle to turn to its "flipped" position of 270 degrees. 

The "flip1" variable must be -1 for the turtle to move to its "flipped" position.

Turn Left 90 Degrees

 The "Turn Left 90 Degrees" block is coded exactly the same way but with the turtle will turn to the left.

Using this code, I recreated seventy of Waclaw Szpakowski's works in TurtleStitch.

I did this by copying each of Waclaw's Szpakowski's designs and placing it in the Silhouette software. Whereby I was able to resize and overlay a series of blocks on top of each design.  I was then able to determine the numbers of steps needed to complete each line by counting the blocks and coding it using the TurtleStitch software.

In the following photos, I exported the coded images as an SVG in TurtleStitch and copied it into the Silhouette Cameo software where I was able to print the designs using an ink pen and Foil Quill on 65 lb. cardstock.

Ink Pen

Ink Pen

Ink Pen

In the example above,  I used a Foil Quill Heat Pen with the Silhouette Cameo. A metallic film is placed on top of the cardstock and the Foil Quill Heat Pen fuses the metallic foil onto the cardstock wherever the pen tracks the design.

Monday, May 30, 2022

A Rubber Band Pop-Up Sliceform Flower

A Rubber Band Pop-Up Sliceform Flower 

Bottom view.  

Notice that there is a diagonal with a rubber band in the base.  The rubber band is stretched to allow the flower to lie flat. This flower is suitable for mailing and will fit into an A7 envelope. I made two versions of the flower.  Many more versions can be made by altering the combinations of the petals, leaves and the insects that I have included in the files. 

Here is the PDF. I used 65 lb card stock and a one inch rubber band for the pop-up base. (The rubber band is similar to the rubber band that is used on a Rainbow Loom.) 

Here is the .Studio file.

Here is the SVG.

Slide the two largest semicircles together to form the center slices.

Insert all of the upward facing slices.

Complete the center of the flower by inserting all of the downward facing slices.

Fold one of the pop-up bases in half and apply glue to one side of the base. Repeat for the other pop-up base.

Crease the base as shown above.  Glue the smaller tab together and insert the one inch rubber band. Apply glue to the inside diagonal as shown above.

Form the base as shown above.

Insert the other side of the rubber band so that the diagonal can now lie flat.

Apply glue to the pop-up columns and attach to the corners opposite the diagonal.

Apply glue to the pop-up column posts and adhere to the inside of the sliceform.

In the center of the above photo, the post and the sliceform can be seen where it was attached.

Glue the petals to the sides of the base. The petals can be creased upwards or downwards depending on your preference.

Repeat gluing the petals to the sides of the base.  Adhere the insect or the butterfly.  The butterfly has a tab which needs to be attached to the butterfly and the petal.

After the petals have been glued on, glue the leaf and the base wrap. Notice that the base has been depressed and it is flat.  By making the base flat, it makes it easier to glue and align the base wrap.

Bottom view

Top View

Top view with butterfly.

Bottom view 

I am posting these pop-up sliceform flowers on Memorial Day in remembrance to all the people who have served in the military including my father who served in World War II in the Philippines. To the great men and women of our armed forces, thank you for fighting for our freedom and sacrificing so much for us.

Monday, May 2, 2022

A Sea Creature Pop-Up Snow Globe

This sea creature snow globe uses the rubber band pop-up mechanism that was created in my previous two blog postings. and

There are plenty of sea creatures packed into this snow globe. There is an octopus, a sea horse, a star fish, a sea turtle and coral inside this snow globe. This pop-up will delight children and adults alike.

The back view looks great too!
This sea creature pop-up snow globe lies flat and will fit into an A7 envelope. The rubber band mechanism inside the base will deploy and the snow globe will expand when it is taken out of the envelope.

Here is the PDF. I used Neenah brand 65 lb metallic card stock from Office Depot. A one inch rubber band is required for the pop-up base. (The rubber band is similar to the rubber band that is used on a Rainbow Loom.)

Here is the .Studio file.

Here is the SVG.

Make the slice form sphere. The slice form sphere directions are in this blog posting, 

Make and assemble the slice form sphere base and rubber band base using the directions in this blog posting.

Wrap and glue the base with pretty paper to complete the pop-up sea creature snow globe. 

Sunday, April 24, 2022

A STEM Project: Resizing a Slice Form Sphere

After making the slice form sphere with a pop-up base.  I tried to insert the pop-up into an A7 envelope.  Much to my disappointment, it was too big by 1/4 inch.  Typically, slice forms can not be resized because the slits in the paper need to accommodate the thickness of a piece of paper. 

The slice form was 1/4 inch too big for the A7 envelope.

I looked at the original size of the slits of the slice form sphere and they were .02 inches wide.  The slits in the slice form base were a little smaller at .17 inches wide. I know that the slits can be as tight as .15 inches wide (not optimal but it does work). Resizing the design would be easy with the Silhouette software, so there is no harm in trying this method before redoing the slice form models. To resize the 4 inch slice form sphere to a 3 3/4 inch sphere.  I calculated that I needed a .9375 scaling factor.

3.75/4 = .9375 scaling factor

In the Edit function, select all.  This will put a bounding box around the entire design.

In the Transform function, under scale, use a .9375 scaling factor.  After the entire design was rescaled, I determined the smallest slit was .17 inches and that is an acceptable slit width. 

The point of this blog posting is that while conventions might indicate that something won't work.  Try it anyway, I had nothing to lose except a few minutes of time.  This is one of the reasons I like designing with paper.  It is quick to experiment and the cost of the paper is reasonable.  I typically make the same model over and over again until I am satisfied with the results. I have high standards and will not concede. In this case, it was a simple solution of rescaling the design.

My resized pop-up slice form now fits in an A7 envelope.

Saturday, April 23, 2022

A STEM Project: Making a Pop-Up Base For A Slice Form Sphere

I have always wanted to create a pop-up mechanism for my slice forms.   I think this rubber band mechanism achieves my goal. 

Slice form base with a rubber band pop-up base attached

When the side is depressed, the base lies flat.

In this blog posting, I will make a pop-up base for the slice form sphere that was created in a previous blog posting, The entire design, slice form sphere, slice form base and pop-up base can lay flat as a two dimensional object and then it can be expanded into a three dimensional model with the help of a rubber band.

Here is the PDF.  I used 65 lb. cardstock. A one inch rubber band is required for the pop-up base. (The rubber band is similar to the rubber band that is used on a Rainbow Loom.) 

Here is the .Studio file.

The slice form sphere file and directions are in this blog posting, 

The pop-up base consists of a two inch square with a diagonal going across the square.  This diagonal is cut into two lengths which equal the entire length of the diagonal.  A rubber band keeps the two diagonal pieces together. 

The rubber band expands when the base is pressed down at the corner that is opposite to the diagonal.  When the side pressure is released, the expansion tension on the rubber band is released and the rubber band returns to its natural state  This phenomenon allows the slice form to deploy and become three dimensional.

Here are the calculations that I used to create the paper strip.  One strip creates a right triangle.  Two strips create a square with a diagonal in the middle. 
Pop-up base Calculations

Make the Pop-Up Rubber Band Base

There are two sides to the rubber band base. Each side is folded over to create a two-ply base. Crease the paper as shown above.  

Glue the folded sides together.

Glue the tabs together. Make sure that the slits are going in the same direction.

Slide the one inch rubber band into the slit and align the rubber band with the round hole in the base of the slit.

Apply glue to the diagonal as shown above.

Adhere the other side of the diagonal.  Make sure that the slits on the long side of the diagonal align.

Slide the rubber band onto the slit on the long side of the diagonal.

This rubber band base can now fold flat when the corner opposite the diagonal is pushed down.

Make the pop-up columns by creasing the paper into a square and gluing the tab together. Repeat for the other column.

Apply glue to the side of the column that has a tab. Attach this side to the right angle that is opposite the diagonal.  Repeat for the other column.

Make the Slice Form Base

Arrange all of the slices by size.  Take the largest slice and slide them together as shown above.

Continue sliding the pieces with the upward facing slits onto this assembly.  

Slice the downward facing slits onto the assembly to finish making the slice form base.

Make the Slice Form Sphere Using the directions from this previous blog posting

Assemble the Three Parts of the Pop-Up 

Apply glue to the column tabs as shown above.

Slide the slice form base onto the pop-up base.  The tabs of the columns will be glued inside the slice form  base. Flatten the base so that the glue adheres correctly. 

The slice form base is now attached at two spots.  In the photo above, it is attached on the left side.  The right side is not attached as this allows the slice form to fold flat.

Top View

Apply glue to the tabs on the center slice of the slice form base.

Adhere the tabs to the center slice of the slice form sphere. Wait for the glue to dry before collapsing the pop-up.

The pop-up will fold flat.

Rubber Band Pop-Up Base with a Slice Form Sphere