Business Card - Platonic Polyhedrons / Executive Desk Puzzle

Dedicated to the West Coast Origami Guild (WCOG) where I probably learned one or two of these polyhedrons. "Long may they fold! May they never fold!"

Grab your business cards and follow along to create a set of "customized" Platonic Solids and assorted polyhedrons, to proudly display on your desk.

By some platonic coincidence, the 3.5" x 2" size of standard business cards is nearly the perfect dimension for the simple construction of equilateral triangles, rhombuses and 'flaps' that can interlock to create an astounding variety of amazing polyhedrons. The true "Platonic Ratio" for a rectangle needed to get "perfect" equilateral triangles using the following method is √3:1 (less than 2% variance from Business Card dimensions – but you'll hardly notice the difference!).

Material (a 100% paper based instructable):

1. Business Cards
2. No scissors, no knife, no glue, no tape. Nothing else !!

1. Right-hand triangle unit:
2. Take a business card with the 'business side' (printing) away from you, and fold point A to point C, creasing the fold well (see diagram). Leave the card folded this way!
3. Take the two remaining edges of paper on the side (where you only have 'one' layer of paper) and fold them over the 'double' layer in the middle, and crease the folds well.Those two remaining single layer edges folded this way effectively 'hug' the center double layers, holding it flat.The center double layer should look like an equilateral triangle. (see diagram).
4. Open the folded cards just a 'bit' – do not open the creases much!
5. Left-hand triangle unit
6. Repeat steps 'a' thru 'c' above on another business card, but in step 'a' fold from point B to D instead of A to C!

1. Tetrahedron (1 Right-hand triangle units + 1 Left-hand triangle unit, difficulty 1/10 ): The 'simplest' Platonic Solid has four equilateral triangle faces. I highly recommend you start by creating this solid first, which will greatly help you understand the construction technique, which becomes more complex with the higher order polyhedrons. Please don't confuse the Greek tetrahedron with their Egyptian cousin the pyramid!
2. Imagine each folded card like the beak of a bird (e.g. < ). Now place the two bird beaks in front of each other, 'face to face' (e.g. < > ) and slowly, bring them closer together as if they were going to kiss… lips lightly parted, each head tilting slightly opposite of the other, hearts racing in breathless anticipation, a frisson tingling up their spine. . . BUT I DIGRESS!!! I assure you the relationship between the two cards is completely 'Platonic' !!
3. Fit the two pieces together so that the triangle faces join to form the tetrahedron, with the 'hugging' flaps on the outside of the central tetrahedral shape – notice the flaps from each card 'hug' the triangles from the other card, and vice versa!This is akin to interlocking the flaps of a cardboard box, where each flap keeps its neighbor flap down, and is likewise held down by its neighbor flap on the other side.
4. No glue is needed, the only thing holding the form together is the reciprocating tension of the card pieces and sheer will. Now go amaze your friends by arming and dis-arming this model right in front of their eyes!!

1. Octahedron (4 Right-hand triangle units - OR 4 Left-hand triangle units, , difficulty 5/10 ):An Eight sided solid, each side is an equilateral triangle. It's a multi-faceted solid, a real diamond in the rough. . .
2. Place the slightly opened cards next to each other, vertically, and in parallel – keeping the same 'orientation' of the printed sections will make the results all the more impressive. Now, fit the four pieces together so that the triangle faces join to form the octahedron, with the 'hugging' flaps on the outside of the octahedron. Notice the flaps from each card 'hug' the triangles from the card next to it, and vice versa! Again, this is similar to the concept of interlocking the flaps of a cardboard box to keep it closed. (note the final model in the picture has ONE differently colored business card, to more clearly display the orientation).
3. TIP – if you are having difficulty keeping the cards in the structure in place while you are trying to put it together, try placing a scrunched up Kleenex on the inside of the structure – this gives some internal support to the structure to counteract the external pressure you place on it while joining the pieces together with your big clumsy hands.
4. To prove that the only thing holding the form together is the reciprocating tension of the card pieces and the force of sheer will, I will hold the octahedron between my finger and thumb and gently blow on one side, and the flaps will catch the breeze and gently spin in a most hypnotic manner – then I will bring my thumb and finger closer together, causing the shape to burst apart, and fall into pieces on the floor. Have fun assembling, twirling, and dis-assembling your customized octahedron!

1. Icosahedron ( 5 Right-handed AND 5 Left-handed units, difficulty 10/10): This solid has 20 equilateral triangle shaped sides! If you thought the Octahedron was tricky, you're going to really hate this one – but the result is sooooo satisfying!!
2. Start constructing the solid by putting two similarly creased cards together, interlocking one flap from one card on top of a free edge of a triangle of the other card.Each card has 2 triangles, so putting these two cards together results in a structure with 4 triangles meeting at a point – but note that each 'vertex' of the icosahedron is formed where FIVE triangle faces meet. So the only type of card that could join with these two cards to equal five triangles would be a card of the "other" crease pattern, otherwise the interlocking flaps and triangles would not line up! Always select which crease pattern card to use based on what next fits into the existing 'flap' / 'triangle' configuration.
3. As with construction of previous Platonic solids, keep the 'flaps' on the outside of the body, and the 'flaps' from one card will serve to secure the 'triangle' face of its neighbor card, and that neighbor cards 'flaps' will serve to secure the 'triangle' face of the next neighbor card, and so on, and so forth.This is akin to interlocking the flaps of a cardboard box together… in the 4^th dimension… while playing a tuba… during a thunderstorm!!
4. TIP – if you are having difficulty keeping the cards structure in place while you are trying to put it together, try placing 3 or 4 scrunched up Kleenex on the inside of the structure – this gives some internal support to the structure to counteract the external pressure you place on it while joining the pieces together with your clumsy mitts. Additionally, if you continue to have difficulty putting the pieces together, you now have Kleenex handy to cry into and wipe your tears.
5. TIP – I have put this model together multiple times, and I have never had to 'plan' which type of creased card (Right-handed or Left-handed) to use; I just look at what next fits into the existing 'flap' / 'triangle' configuration, and somehow the laws of nature always allow that the final finished model is completely interlocking, with 'flaps' lining up with a free side of another triangle.
6. To prove that the only thing holding the form together is the reciprocating tension of the card pieces and sheer force of will, I will apply increasing pressure on the sides of the icosahydrohedron until it explodes into a million (actually 10) pieces.

1. Cube (6 cards, difficulty 3/10): This Platonic Solid has 6 square sides. It is a superior shape because it has all the "right" angles…
2. Take two business cards and put their non-printed sides together, with the cards perpendicular to each other, so they form as perfect of an 'X' as you can.
3. Take the four remaining edges of paper on the sides (where you only have 'one' layer of paper) and fold each one of them over the 'double' layer in the middle, and crease the folds well. Thesesingle layer 'flaps' folded this way effectively 'hug' the center double layers, holding them together. The central double layer should look like a square!
4. Repeat steps 'a' & 'b' on two more business cards.
5. Repeat step 'c' on two more business cards.
6. Open the folded cards just a 'bit' and separate them – do not open the creases too much!
7. Place the six pieces together so that the square faces form a cube, weaving the 'hugging' flaps to the outside of the adjoining square faces, so that the flaps from one card 'hug' the square faces of two other cards – and vice, vice, vice versa! Again, this is similar to the concept of interlocking the flaps of a cardboard box to keep it closed. (see diagram)

1. Right-hand Rhombus form:
2. Take a business card with the 'business side' (printing) away from you, and fold point D inwards along a line that passes through point A, so that side AD now lies on the imaginary line thar runs from point A to C (see diagram). Leave the card folded this way!
3. Now fold point B in a similar way inwards along a line that passes through point C, so that side BC now lies on the imaginary line that runs from A to C (see diagram).Note that point D and point B should be nearly touching in the center of the resulting rhombus shape. Leave the card folded this way.
4. Left-hand Rhombus form:
5. Take a business card with the 'business side' (printing) away from you, and fold point A inwards along a line that passes through point D, so that side AD now lies on the imaginary line thar runs from point D to B (see diagram). Leave the card folded this way!
6. Now fold point C in a similar way inwards along a line that passes through point B, so that side BC now lies on the imaginary line that runs from B to D (see diagram).Note that point A and point C should be nearly touching in the center of the resulting rhombus shape. Leave card folded this way.

1. Rhombohedron (3 Right-hand Rhombus AND 3 Left-hand Rhombus forms, difficulty 6/10): it's basically just like the cube above, except that instead of 6 square faces, it has 6 Rhombus faces.
2. Like the Cube, this Rhombohedron is constructed with similar interlocking flaps.
3. TIP – Approach that worked for me is to put 3 Right-hand Rhombus together so they meet at a point (use the scrunched up Kleenex trick to keep internal pressure if you have difficulty), then add the Left-hand Rhombus one at a time, forming an "opposite" cone to the 3 Right-hand Rhombuses…
4. The Rhombohedron is held together only by reciprocating tension of the cards!

1. Rhombic Dodecahedron (12 Right-hand Rhombus OR 12 Left-hand Rhombus forms, difficulty 11+/10): Putting the 3 rhombus points together to form the Rhombohedron, I got to thinking, what would happen if I put 4 rhombus points together… I did just that, and kept adding rhombus forms to the resulting shape based on where flaps & triangles met… and this structure built itself! It has 12 congruent Rhombic faces, and if you thought the icosohedron was difficult, you're going to despise this one! But the result is soooooo cool!
2. Put 4 Right-hand Rhombus together so they meet at a point (use the scrunched up Kleenex trick to keep up the internal pressure and keep the units from falling in on themselves during construction). Then add Rhombus forms one at a time, interweaving the flaps with the Rhombuses, so that the 'flaps' of one Rhombus 'hug' the open rhombus side of it's neighbor, and vice, vice, vice versa!!
3. To give a better understanding of how it's put together I made one out of two different colored 'business cards'. It really breaks my heart to prove that it's only held together by the reciprocating tension of the cards (see video).

1. Right-hand Concave unit:
2. Take a business card with the 'business side' (printing) away from you, and fold point A to point C, creasing the fold well (see diagram). Leave the card folded this way!
3. Take the two remaining edges of paper on the side (where you only have 'one' layer of paper) and fold them over the 'double' layer in the middle, and crease the folds well.Those two remaining single layer edges folded this way effectively 'hug' the center double layers, holding it flat.The center double layer should look like an equilateral triangle. (see diagram).
4. Open the folded card and flip it over with the printing facing you.
5. Fold point B to point D, creasing well. This crease should NOT lie on top of the crease made in step 'a', but should form an X with the crease from step 'a'. Leave the card folded this way.
6. Take the two remaining edges of paper on the side (where you only have 'one' layer of paper) and fold them over the 'double' layer in the middle, and crease the folds well.Those two remaining single layer edges folded this way effectively 'hug' the center triangular double layer flat.
7. Open the card with the 'blank' side facing you, in 'landscape' mode.
8. Fold the bottom half of the card up to the top half and crease well.
9. Open the card and coax all existing crease lines to fold equally, as much as they can .The resulting folded form should Not lie flat, and has four sections running along the length of the card, which we will label, in order:
10. A triangular 'nodule' (for lack of a better word!)
11. A tetrahedral triangular section
12. Another tetrahedral triangular section
13. Another triangular 'nodule'
14. Left-hand Concave unit
15. Repeat steps 'a' thru 'g' above on another business card, but in step 'a' fold from point B to D instead of A to C, and in step 'd' fold from point A to C instead of B to D.

1. Octahemioctahedron (4 Right-hand concave units OR 4 Left-hand concave units, difficulty 5/10): A cool name, weird concave sections, a pleasing symmetry, and the result is really satisfying.
2. On the backside view of the unit, notice that each tetrahedral triangular section has a 'slot' where two pieces of card come together. Place the four folded cards next to each other, vertically, and in parallel – keeping the same 'orientation' of the printed sections will make the results all the more impressive. Put the triangular 'nodule' from one card (i) into the tetrahedral 'slot' (ii) from its neighbor card! Keeping each unit folded along it's creases (and not splayed out) will help. Do this for all of the triangular 'nodules' (i) into their neighbor card tetrahedral 'slot' (ii).
3. Now comes the fun part. Put the 'other' triangular 'nodule' (iv) into the tetrahedral 'slot' (iii) from its neighbor card! And now do this for all of the triangular 'nodules' (iv) into their neighbor card tetrahedral 'slot' (iii). This is akin to closing a cardboard box by 'interlocking' the flaps – only a bit tricker since there's not a lot of freedom of movement in the 'nodules' nor the tetrahedral triangular section!!
4. TIP – run the video showing Octahemioctahedron coming apart in reverse, to get a better idea of how to put it together!
5. BONUS Challenge – try forming the Octahemioctahedron from 2 Right-hand concave units AND 2 Left-hand concave units! It is possible using a different construction approach

1. Triangular Gyrohemibirotunda ( 3 Right-hand concave units AND 3 Left-hand concave units, difficulty 8/10): So I'm putting together the Octahemioctahedron, and gets to thinking, what if I put only 3 of the units together (from Octahemioctahedron step 'a') ?I did, and this unit just built itself!2 concave triangular faces, 6 triangular faces, 6 concave pentagonal faces!
2. Put three Right-hand concave units together, similarly to the octahemioctahedron, locking the nodules into the tetrahedral triangular slot.
3. Put three Left-hand concave units together, similarly to the octahemioctahedron, locking the nodules into the tetrahedral triangular slot.
4. Now weave the loose ends from the Right-hand section into the concave pentagon section of the Left-hand section, and vice, vice, vice versa!
5. The model is held together only by reciprocating tension across the cards.

1. Icosihemidodecahedron (10 Right-hand concave and 10 Left-hand concave units, difficulty 11+/10): I'm a glutton for punishment, so I joined 4 units, then joined 3, now I'm thinking, what if I join 5 of these concave units together? Here goes!
2. Similar to the icosohedron, start by joining together 2 similar crease pattern units together, e.g. the 'nodule' of unit #1 into a 'slot' of unit #2 - note that the 'points' of 4 equilateral triangles should come together at a point.
3. Now, the same crease pattern unit will NOT fit with the 4 points to create the concave pentagon, so use a unit with the 'opposite' crease pattern. Put the 'nodule' of the 'opposite' crease pattern into the slot of unit #1 from step 'a', and put the 'nodule' of unit 2 into the 'slot' of the 'opposite' pattern unit – this creates your first pentagonal concave face.
4. Have Kleenex nearby, not to put into the middle of the polyhedron (since it is concave it has internal structure to keep it from caving into itself), but to wipe your tears when the whole structure falls apart as your putting in the last piece – as happened to me a few times ☹.
5. TIP – I have put this form together a few times, and never need to 'Plan' what pieces will go where, just look at the model and where you see a 'nodule' and a 'slot', use the concave unit that has the 'slot' and 'nodule' in the correct alignment to fit together to make a concave pentagonal face.
6. Nothing is holding the shape together other than the reciprocating tension of the cards. Now have fun assembling and dis-assembling this concave polyhedron.

1. Dodecahedron (12 cards, difficulty 6/10, tediousness 11+/10): OK, OK, OK !! I know there is ONE missing Platonic Solid – but it has 12 pentagonal faces, no triangle or rhombus in the lot! For the sake of completeness, I have devised a way to build the Dodecahedron using only business cards, but it is NOWHERE near as elegant as previous polyhedrons… This model uses the most cards, is the most tedious, is the least stable, and is my least favorite – sorry Plato!
2. Take a business card with the 'business side' (printing) away from you, and do NOT fold it, but put vertical crease marks in the top middle and the center middle, then flatten out the card again.
3. The purpose of crease marks is to mark points that will be used to guide later folds; the creases should NOT make the paper fold or bend; the crease mark just needs to be visible. Making crease guides too hard will have a negative effect on the final model.
4. Now put horizontal crease marks in the center left, center right and center middle.
5. Now just use your fingernail to make a crease mark (folding would be overkill) in the "middle" of the left edge of the left side bottom half. Do the same on the right side.
6. Now use your fingernail again to make a crease mark in the "middle" of the left edge of the top half of the halves you created in step 'd'. Do the same on the right side. All this creasing is to find and mark the point that is 1.25" from the top of the card (card is 2" high).
7. Fold from the point on the left side that is 1.25" from the top to the top 'middle' crease mark. Do the same for the right side.
8. Fold the left flap so that the point that was 1.25" from the top lies directly on the folded edge from step 'f' and also so that the center of the left flap lies directly on top of where the horizontal and vertical center crease guides cross. Do the same on the right flap. The resulting shape should be a very close approximation of a regular pentagon.
9. Repeat steps 'a' through 'g' on two additional business cards.
10. Arrange the 3 folded cards so that the right flaps are together, and each card is rotated 120 degrees from its neighbor (see diagram). The pentagon sides should nearly be touching.
11. Starting with one pair of the three folded cards, put their pentagon edges that are nearly touching together, so the pentagon edges of this pair are flush together.
12. KEEP this pair of pentagon edges flush throughout this entire step! The flush edge of one of these pairs was created by the fold in step 'f' ; raise that fold 90 degrees so that it aligns with the "right" tab piece from the other card it is flush with (see diagram); this 'right' tab piece should have a small triangular section that sticks out further than the step 'f' fold. Keeping the edges of this pentagon pair flush, fold that small triangular section down and over the step 'f' fold, so that it 'hooks' onto the 'f' step fold – then fold back down the 'f' step fold to its original position where it was at the beginning of this step. This effectively 'locks' the two pieces together!
13. Repeat steps 'j' through 'k' where the other 2 edges meet.
14. See the resulting modular piece of 3 connected cards - **NOTE – this modular piece of 3 connected cards should be curved and should NOT lie flat. If your finished modular piece lies flat, you probably did NOT keep the pentagon edges flush against each other during step 'k'.
15. Repeat steps 'a' to 'm' Three more times ; when complete you should have 4 of these modular pieces, each modular piece consisting of 3 connected cards … see why this is not my favorite Platonic Solid?
16. Start with 2 of the modules and put them together, weaving the 'flaps' to the outside, so that the 'flaps' of one modular piece hold onto a pentagon of the other modular piece, and vice versa.
17. Add a third module, again weaving the 'flaps' to the outside, so that the 'flaps' of one modular piece hold onto the pentagon of an other modular piece, and vice versa.
18. Add the fourth module, in the same manner as the previous were added.
19. TIP - if you are having difficulty keeping the cards in structure in place while you are trying to put it together, try placing one or two scrunched up Kleenex on the inside of the structure – this gives some internal support to the structure to counteract the external pressure you place on it while joining the pieces together with your big clumsy hands. If you continue to have difficulty putting this shape together, you now have Kleenex handy to cry into and dry your tears.

Q. Who invented the Platonic Solids?

A. Technically God – his wondrous creations like crystals, minerals and molecules had these shapes eons prior to being "discovered" by Plato and other Greek philosophers. Who said "God doesn't play dice with the universe" ?!

Q. Who invented Paper?

A. Again, technically God – his wondrous creatures like wasps were creating paper millennium prior to the Chinese 'discovery' of how to create paper.

Q. Are there any other polyhedron that can be formed using these 'units' ?

A. Probably! What would happen if you join triangle units with rhombic units…?! Or fold the rhombic units along its major axis to get two obtuse triangle units…?! What wondrous polyhedrons could be formed…?! I guess I know what my next instructable will be!

Q. Can I use other types of cards, like maybe Pokemon cards, or playing cards to build these solids?

A. For the Cube yes, but for the other solids you will need to be closer to the "Platonic Ratio". Most "standard" playing cards are 3.5" x 2.5", so you would need to trim that Charizard closer to 3.5" x 2"

Q. Can I use glue or tape to hold a model together?

A. Can you un-cry a tear…? Can you catch a dream in a butterfly net… ?

Q. Are there any other Platonic solids?

A. In the 3^rd dimension, no. In the 4^th dimension (and beyond) yes, but good luck folding those !!

Q. Are you related to JuanV165 ?

A. I can neither confirm nor deny that, aside from saying we have a platonic relationship.