# 12-Card Star

This puzzle/sculpture is filled with deep geometric ideas and is an attractive object for students to build and take home.  Using everyday objects like cards engages students and reminds them of the creative playful aspects of mathematics.  This star can be made with ordinary playing cards or with larger cards for greater impact.  Larger size cards are also easier to manipulate and are especially recommended for younger students.

This is a wonderful exercise in introducing students to abstraction in mathematics.  The 12-Card Star is isomorphic to a cube.  The 12 identical cards in the construction correspond structurally to the twelve edges of a cube.   Even young students will begin to understand how mathematicians see objects at deeper levels and look for parallels between the underlying structures of superficially distinct objects.

The construction relates to modular origami and modular kirigami as there are both cuts and folds in the twelve cards.  Students who have had prior experience with origami will enjoy this and be better prepared.

Be warned that this activity is trickier than it looks!  It will be most successful if the teacher has mastered the steps ahead of time so he or she can demonstrate it with ease.

This activity provides rich classroom material for teachers following the Common Core Standards for Mathematical Practice.  This lesson also provides cross-curricular connections to art, architecture, and higher-level strands of mathematics.

# Detailed Instructions

## Time Required:  2 Hours

• 1 Hour Preparation (Part A)
• 15 Minute In-Class Discussion (Part B)
• 45 Minute In-Class Construction (Part C)

## Materials

• 13 decks of cards, preferably giant size (e.g., https://www.dollardays.com/i1776566-wholesale-giant-playing-cards.html)
• Two or more different models of a cube, of which one is ideally an edge model, e.g., Zometool or straws joined by pipe cleaners.
• Metal rulers or access to desks or other flat surfaces with a crisp edge for folding.  (See explanation below.)
Notes:
1. 13 decks of cards are sufficient for 50 students.
2. Inexpensive standard-size cards (making a 14 cm diameter sculpture) can be purchased at a "Dollar Store" but are trickier to work with and will be too difficult for younger students to manipulate.  Giant cards can be purchased in quantity online and result in a more impressive 28 cm sculpture.
3. This activity can be done without a band saw, but it is tedious and time consuming to cut the slots with scissors.  If the school does not have a band saw, it is worth asking around to find a carpenter or woodworker who does have one.
4. A clean, crisp fold is essential to the overall aesthetic of the final result, so it is important to have a straight-edged surface which will serve as a guide for the fold.  Desks often work well as long as the edges are not curved.  Sometimes a cabinet, window sill, or other aspect of the room environment has a suitable edge.  A metal ruler also works well, but some wood or plastic rulers do not have a good edge for folding.  Explore the room and experiment beforehand.

Part A.  Preparation

1. Print and cut out 13 copies of the template that matches the size of your cards.  Using scissors you only need to cut out the rectangle shape. Don't cut the slots in the template.  Here are PDF files you can download for standard US poker cards and for oversized cards. If neither template matches your deck size, you can scale these proportionally before printing.

2. The slots must end up as shown in the above image, and not its mirror image.  So before cutting, you need to determine which side of the package has the face-side of the card up and which side has the back of the card.  If your decks of cards are wrapped in clear plastic, you can tell which side is up by looking through the plastic.  If the deck is in a cardboard box, you will need to open each deck to determine this.  If the box holds the cards tightly, they can be left inside the box for cutting, but if they are loose inside the box, they will need to be taken out and taped into a tight bundle:
• With clear wrapping, keep the cards sealed in the plastic and tape a template face up to the face side of each deck.
• With opaque wrapping in a tight box, look in the box to determine which side is face up and tape the template to the outside of the box on the face-up side.
• With opaque wrapping in a loose box, take each deck out of its box, being careful not to mix up the order of the cards. Place a template face up on the face side of each deck and use Scotch tape to hold each deck together firmly.  The tape should wrap around to the back card creating a brick-like object.
3. Using a thin blade on a band saw, cut slots in each deck on the four solid lines.  Try to be precise so the slot is neither too short nor too long.

4. Keep the decks unopened, so the cards remain in order, until the assembly activity.

Part B.  In-Class Discussion

Before the construction, it is valuable to talk about the geometry of the object the students will make, especially "isomorphism" and rotational symmetry.

Students will be familiar with rotational symmetry.  Using a model, solicit the fact that a cube has four-fold rotational symmetry.  If you turn it 90 degrees about a line through the center of a face, it looks unchanged.  It may be harder for students to see that the cube also has three-fold rotational symmetry about a long (body) diagonal.  A good way to illustrate this is to stand a cube on a corner and rotate it 120 degrees so it appears unchanged.  Finally, the 2-fold symmetry can be demonstrated by rotating it 180 degrees with only one edge resting on the table and the opposite edge directly above it.

Introduce the idea of an "isomorphism."  While the term is not in the standard K-12 math vocabulary, it is an important concept that students can easily understand in a general way and one not to shy away from.  There are various definitions of the term, including some technical mathematical definitions that can be ignored here.  We use the term simply to mean a one-to-one structural correspondence between the parts of two objects.  For example we would say the ten fingers of the hands are "isomorphic" to the ten toes of the feet.  When you visualize hands or feet, you see that the ten digits come in two mirror-image sets of five, with the large ones near the middle.  Based on this understanding of their common structure, there is a natural one-to-one mapping: for each toe, there is an obvious pairing with a particular finger and vice versa. This type of relationship between things that may be superficially different is called an "isomorphism."  Mathematicians look for such parallel structures in everything they see.

After introducing the idea of isomorphism with hands and feet, a good mathematical example is to show two very different models of a cube, e.g., one could be a solid wood cube and the second could be made of marshmallows and toothpicks.  Keeping them parallel, you can point out that they are isomorphic.  A particular corner of the wood cube corresponds to a particular marshmallow of the other cube, etc.  After showing the one-to-one correspondence of the various components of a cube, (vertices, edges, faces), you can then point out that the relationship between those components on the wood cube corresponds exactly to the relationship between the same components on the marshmallow cube.  As an example, an edge touches a face on one cube if the corresponding edge touches the corresponding face of the other cube.  Once students understand this concept, tell them that the object they are going to make is isomorphic to a cube.

Tell students they will be using twelve playing cards to make a structure isomorphic to a cube.  (The thirteenth card is a spare in case something horrible happens.)  Solicit ideas from the students about what those twelve cards might represent on a cube.  What does the cube have twelve of?  Once students realize that the twelve cards will correspond to the twelve edges of a cube, look at a vertex of a cube and discuss how three edges come together at each vertex.  There must be an isomorphic aspect of the card construction---a place where three cards come together.  Similarly, because a face is surrounded by a cycle of four edges, there must be an isomorphic aspect of the construction in which four cards make a cycle.  Tell students that their mental image of a cube should guide them in assembling the card construction.  The end result might not look like a traditional cube, but they will see the one-to-one correspondence or isomorphism.

Part C.  In-Class Construction

Note: The construction will be most successful if the teacher has mastered the steps ahead of time so he or she can demonstrate it with ease.

Note: Steps 1-6 work well with older students, but with younger students you may want to do these steps yourself as part of the preparation.

1. Arrange a long straight surface such as a counter, a pair of tables, or a series of desks to create 13 stations for students to pick up their cards.

2. Unwrap the 13 decks and space them face up in a row along the surface, ideally a couple of feet apart.  When taking the cards out of the packaging, be careful not to mix up their order.

3. Walk along the row and take the top card from each deck, so you have 13 identical cards to demonstrate with (probably jokers).

4. Tell the students that they will walk along the line from left to right taking the top card off of each deck, as you did.  Their goal is to end up with 13 identical cards.  The first person might get all the Aces of Spades, for example.  Warn them that sometimes two cards stick together and they accidentally get a card intended for the person behind them.  A good way to prevent that is for students to look at the top card on the pile left behind and make sure it is always consistent, e.g., the 2 of Spades.  If it isn't, they should check if they accidentally picked up a card of the person behind them.  At the end, they should count that they have 13 identical cards.  Tell students not to play with the cards until you give the next instructions.

5. Once they understand the whole process, ask the students to line up, collect their cards, and return to their seats. Allow for enough time to do this without confusion.

6. Ask students to double-check that they each have 13 identical cards.  Resolve any possible mix-ups.  [You can decide to allow students to trade entire decks with each other if they prefer a different card.]  Again remind students not to play with the cards yet.

7. Draw student's attention to the structure of each card.  There are two long slots and two short slots.  The two ends are the same, so you can turn a card 180 degrees and it looks unchanged, i.e., it has 2-fold rotational symmetry.  The long slots lie along a common diagonal, while the other diagonal connects the two value symbols, e.g., the A's of an Ace.

8. Remind students again not to start folding until they have seen all the instructions.  Show students how they will fold along the diagonal that doesn't have any slots. This means the fold will go through the two value symbols, NOT where the slots are (which is very tempting, you will discover...)  Point out that you will fold the back to the back so the face of the card is on the outside.  Use the edge of a ruler or the edge of a desk as a guide to start the fold.  Tell students to position the card over the straight edge making sure the corners of the card are aligned with the edge.  Push down with your thumbs to start a clean, crisp fold.  Then continue the fold by flattening the card in half against the desk surface.  Rub a fingernail or a ruler along the fold to make a hard crease.  The card will naturally spring back open to roughly 60 degrees. No additional bending or creasing is necessary.

9. Once they understand the folding process, ask students to fold their own cards.  Walk around and check students' work.  The folded cards are all identical, with the same four slots and the same crease.

10. Tell students that they will now begin the assembly.  Remind them there are long slots and short slots.  Instruct students that they will always join a long slot to a short slot---never a short to a short; never a long to a long.  Build a three-card module and show it to the students.  (See Step 11 for detailed instructions.)  Explain that they will put together three cards to make this module with three-fold rotational symmetry.  Point out that this corresponds to a vertex of a cube where three edges meet.

If this is done correctly, the cards will lock together.  If you do not master this step, the structure will fall apart.  The back of the module shows a triangle in its center, like this:

Allow students time to discover how three cards come together correctly.  Let them puzzle it out for a while.  Some will find it very difficult.  The students who solve this quickly can help their neighbors.

11. For students who need more detailed instructions, slide a short slot into a long slot, as shown. The lengths work out so the edge of one card just comes up to the crease of the other card. The third card slides in to make two more of this same kind of joint. This long-slot-to-short-slot joint is repeated throughout the construction and happens three times in a three-way module, connecting the three cards in a cyclic fashion: A to B, B to C, and C to A, as shown above.

This video shows the details:

12. Ask students to make a second three-way module in the same way.  Tell them that the easiest method of assembly is NOT to make a third three-card module, but to keep adding one card at a time from this point on.  The first two modules are joined together and secured with a new card, making a new three-way joint.  At this stage, seven cards have been used, as in this photo:

There is a chain of four cards that can be closed into a cycle using another loose card.  This cycle of four makes a square that corresponds to a face of the underlying cube.  These eight cards look like this:

13. From here on, let students add one card at a time, following the same patterns until twelve cards are used.  The cards can be added in any order.  When walking around helping students, check for the following:
• Make sure there is no cycle of five.
• Make sure the slotted corners of the cards all end up on the inside.
• Make sure to keep checking on the inside that the 3-way joints are correctly made.
Students who finish first should be encouraged to help others.

Warn students not to be upset if it sometimes falls apart during assembly. The cards are slippery, but everything locks together securely when the twelfth piece is inserted.

14. When complete, you will see eight 3-way cycles as in the image above, representing the eight vertices of a cube.  There are also six 4-way cycles, representing the faces of a cube.  Point out the symmetry axes.  You can ask students to find the 2-fold axes, which correspond to the line from the center of a cube's edge to the center of the opposite edge.

## Possible Extensions

Larger Structures.  Observe how two of the stars can join together by placing a 4-way pinwheel of one on a 4-way pinwheel of another.  By thinking about the XYZ axis directions, you can join eight stars together in this way as the vertices of a larger cube.  It can be extended into a larger cubical array. A full set of 52 stars might be made into a 13x4 array with rows and columns corresponding to the suits and values of the deck.

What is the Angle?  The fold in any loose card is like a door hinge that could be set at any angle. During the assembly process, the cards join in a way that forces the fold angle to a particular value without you having to plan for it.  Can you determine what the fold angle is in the completed star?   [Ans: By looking down along a 3-fold axis, you can see three cards on edge which form a large equilateral triangle, so the fold angle finds a value of 60 degrees.]

Combinatorial Puzzle. After mastering the geometry, you may choose to add another level to the puzzle.  Using the Ace, King, and Queen of each suit, discover how to assemble them so each 3-way vertex consists of an Ace, a King, and a Queen from three different suits, and each 4-way vertex displays "two pairs" and all four suits.

Other Isomorphisms.  One can explore the idea of isomorphisms with many other examples. For older students familiar with exponentiation and logarithms, one could point out the isomorphism between multiplication of numbers and addition of exponents.

[Future Idea: Something related to the fact that the form derives from the third stellation of the rhombic dodecahedron...]