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Pyramid Vertex Module (PVM) — folding instructions
These are folding instructions for the Pyramid Vertex Module (PVM) designed by Michał Kosmulski.
The Pyramid Vertex Module is actually a family of units. It consists of vertex units which look like trigonal pyramids (hence the name) and connector units which are used to connect the vertex units. Different combinations of vertex units and connector units result in different models, sometimes as different as to not be obvious that they are built from very similar modules.
Vertex modules are very paper-efficient. In most places, there is only a single layer of paper, and the finished units are quite large relative to paper size. They can be folded from standard paper, but using somewhat thicker paper can make them more sturdy. Connector units, in contrast, have quite many layers of paper and will not work well if folded from paper that is too thick.
This module was designed and published in January 2015 by Michał Kosmulski. Of course you are free (and encouraged) to use it for folding your models and to use it as an inspiration when inventing your own modules.
After publishing this unit, I learned about a similar design by Miyuki Kawamura. I don’t have access to instructions, but both the connectors and the vertex units seem to be slightly different than mine.
The pictures (diagrams) below are available under a Creative Commons license.
Finished module and sample usage
Folding the Pyramid Vertex Module
Vertex unit (regular)
1. Start with a square piece of paper. Crease both diagonals.
2. Crease from the midpoint of one edge to the center point where diagonals cross.
3. Collapse along the creases to form a trigonal pyramid (view from the bottom and the top). The crease will open by itself, but this is not a problem since after attaching connector modules, the vertex module will be held in place as seen in the last picture.
Vertex unit (sunken)
1. Start with step 2 of the regular vertex unit. Make four creases parallel to the sides of the square by bringing the midpoint of each side to the point where diagonals cross (the sheet's midpoint). Only fold the part of the crease which lies between the two diagonals: the parts directly adjacent to the paper's edge only need to be creased for certain models, e.g. the cuboctahedron. Paper will bend near the edges, but if you don't crease it, these parts can be smoothened out later for a cleaner appearance.
2. Fold the flap like in step 3 of the regular vertex unit, but preserving creases you just folded.
3. The end result looks like the regular vertex unit but has a second, smaller pyramid in the middle of the larger one, pointing in the opposite direction. The middle pyramid holds the unit together, so the sunken unit keeps its shape before attaching connector units much better than the regular variant.
1. Start with a square piece of paper. Crease both diagonals and crease in half along both sides of the paper.
2. Fold the preliminary base.
3. From now on, treat the folded preliminary base as if it were just a square sheet of paper. Crease along both diagonals. Find the corner of the small square under which all layers of paper meet and make two creases connecting the adjacent sides of the square with the point where diagonals cross.
4. Fold along the creases, forming a trigonal pyramid and collapsing a flap inside. This step is very similar to folding the vertex unit, but in this case the crease must go through the middle of one of the pyramid’s faces and not along an edge (see the picture of finished connector in next step).
5. Make a crease on the flap to prevent the pyramid from unfolding. Since the connectors hold the whole model together, they need to be sturdy. Notice that one side of the pyramid is different from two others and contains a large crease.
6. This connector binds vertex units in that the are inserted into the connector’s pockets. There are three pockets, one on each of the pyramid’s edges.
1. Start with a square piece of paper. Pleat into fourths along both sides of the paper.
2. Fold along one one-fourth crease.
3. Fold along one perpendicular one-fourth crease, creating a triangular flap where the creases meet.
4. Squash fold the flap.
5. Repeat in the other three corners of the sheet.
6. Turn over to the other side.
7. After previous step, the model looks like a small square. Fold the preliminary base from this square, then space the four flaps so they point in four roughly perpendicular directions (the base will not be flat anymore).
8. In order to connect vertex units with the octahedral connector, move the vertex units’ corners into the pockets located on the four sides of the connector unit. Pockets do not have solid internal walls (there is a slit along each one on the inside) but they can still hold the vertex units quite well.
Octahedral connector for inverted assembly
1. Start with step 6 of the regular octahedral connector. From the small square fold the waterbomb base. Spread the four corners apart so they point in four roughly perpendicular directions (the base will not be flat any more).
2. In this step, the model has to be turned inside-out in a way. Grab the four corners and pull them together into one point above the surface on which the model is lying while holding the center point. The image to the right shows the connector from the other side.
3. In order to connect vertex units with the octahedral connector for reverse assembly, move the vertex units’ corners into the pockets located on the four corners of the connector unit. The for free corners of the connector should meet in one point once you connect all four vertex units. The four small flaps should fold into the spaces along the edges of vertex units.
1. Start with a square piece of paper. Pleat into fourths along both directions.
2. This is the beginning of a squash fold.
3. Squash fold as seen in the image.
4. Repeat the squash fold three more times.
5. Reverse fold the two triangles on the left and on the right. The result is quite many layers of paper which will be able to squeeze and hold other modules later on.
6. Fold in half along the middle crease.
7. Vertex units go into the two pockets located at the ends of the connector unit. Due to the many layers of paper found inside, connectors hold the vertex units quite well even though the pockets have slits on the inside which make them weaker.
Big edge connector
1. Start with step 4 of the regular edge connector, fold in half along the long axis and you’re done. This variant is larger but creates a somewhat weaker link than the regular edge unit.
Reversed edge connector
1. Start with a square piece of paper. Crease both diagonals.
2. Fold three corners to the square’s center.
3. Fold in half on both sides along the axis of summetry.
4. Crease by folding the left edge to the closest corner of the small square. Also, crease the line going through that corner.
5. Crease diagonals of the two small square areas. The creases should be mountain folds in the left square and valley folds in the right one.
6. Fold along the boundary of the two small squares.
7. Tuck the flap into the pocket. The unit looks like a square at this point.
8. Form the waterbomb base from the small square. The square has one solid edge and three edges with pockets. The solid edge and one edge with pocket should be the straight edges in the waterbomb base while two sides with pockets should be used for the parts of the base which are creased inside.
9. Vertex units can be connected by inserting their corners into the two pockets in the connector unit.