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Folding instructions:
Instructions and guide to the "delta-hex" modular unit, as well as the type of polyhedra it can construct.
Preparation/supplies
Preparation/supplies
These units are folded on hexagonal paper, which can be cut from square or rectangular paper. There are numerous methods for cutting out hexagons, such as marking the shape with a protractor or 30-60 degree triangle ruler; printing or using a hexagon template; folding the paper and using halfway marks for guidelines (best applied to A4/B5 style paper), etc. The number of sheets depends on which polyhedron you want to make.
Kami (origami paper) or biotope is a good starting paper. Thicker paper like tant, kraft, or sketch/printer paper may make hexagon shaping more difficult/less precise, but can be worked with (a bone folder will be a significant help). Cutting hexagons from a 15cm square leads to triangular units around 6cm a side, and a hexagon of a decent size for multiple shaping styles. For smaller units, tweezers will probably be useful for hexagon shaping.
Glue is not technically required to secure units together; however, depending on the paper size/type, central hexagon styling, and deltahedron shape, it may be safer and sturdier to apply glue to the flaps.
Folding the raw unit
Folding the raw unit
All units follow the same setup. After you finish the raw unit, the final shaping steps are up to you depending on deltahedron construction and personal preferences.
1: 8x8x8 triangle grid
You will need to fold an 8x8x8 triangle grid onto the hexagon (8 equal divisions between each edge of the paper).
Refer to Madonna Yoder's gridding video tutorial (skip to ~10:00 for the hexagon grids) if you are unfamiliar with this process.
An 8x8x8 triangle grid.
2: Closed hexagon twist
Fold a closed hexagon twist in the center of the grid.
Refer to Twist Database for additional diagrams, animations, and video tutorial if you've never folded a closed hexagon twist.
Diagram showing the fold lines for a closed hexagon twist.
A closed hexagon twist turns the paper into a star shape.
3: Triangular unit
Flip the model over. Starting at one triangular tip, fold in the edges in an orderly sequence. Every other star tip will get an additional tuck to create the three points of the triangular unit.
Diagram showing the fold lines for setting up the triangular edges of the unit and a suggested order.
1) Fold the edge in, pulling in the star tip.
2) Fold the second edge in, leaving the star tip alone. This doesn't resolve neatly but moves towards step 3.
3) Fold the third edge in, pulling in the star tip.
4) Fold the fourth edge in, leaving the star tip alone. This doesn't resolve neatly but moves towards step 5.
5) Fold the fifth edge in, pulling in the star tip.
6) Fold the sixth edge in, tucking it under the first edge (partially unfold the first edge as needed to achieve this.)
Completed raw unit
Flip the unit over. This is the completed raw unit. The central hexagon can be further shaped in a number of different ways. There are also currently 3 stems that can be converted to either flaps or pockets in order to connect multiple units together.
A completed raw unit. There are stems at the top center, bottom left, and bottom right that can be converted into flaps or pockets. The central hexagon twist can be shaped in a variety of ways.
Flap/Pocket setup
Flap/Pocket setup
A raw unit can convert into several different flap/pocket setups: 3 flaps/0 pockets; 2 flaps/1 pocket; 1 flap/2 pockets; 0 flaps/3 pockets.
I recommend a 50/50 mixure of 2 flaps/1 pocket and 1 flap/2 pockets units. This means all units have at least 1 flap and 1 pocket for stronger connections across units. You should be able to pair 2f/1p and 1f/2p units together when arranging units to create polyhedra - keep the pocket and flap orientations on opposite edges, and the combination becomes similar to a sonobe or related modular unit.
A 1f/2p unit connecting to a 2f/1p unit. Note that the flaps and pockets are on opposite edges of the combination.
Creating a flap
Converting a stem to a flap is simple - tuck in the tip of the stem.
1) Flip the unit to the back side. Open up the top layer of the stem.
2) Tuck the stem's tip to the inside.
3) Close the top layer back down.
Creating a pocket
Converting a stem to a pocket is a little trickier as it involves a fairly tight tuck.
1) Flip the unit to the back side. Pull the entire stem down towards the center.
2) Fold/unfold the tip.
3) Fold the stem to the right..
4) Tuck the stem's tip under the top layer (it will be a tight fit.)
Back side view of the pocket.
Looking at the pocket directly.
Central hexagon shaping
Central hexagon shaping
The central hexagon twist can be shaped in a variety of ways. Most of these variations are pulled from designs by Shuzo Fujimoto from his Twist Origami and Invitation to Creative Playing with Origami books.
Pinwheel/Windmill (風車)
One of the simplest variants.
With practice, you can skip most of the precrease step.
An example of a pinwheel unit (2 flaps/1 pocket).
1) Precrease the hexagon by folding each tip to the center.
2) Start folding in the tips of the hexagon to the center in an orderly sequence, with each additional fold going on top of previous fold. To get the last fold done, you will want to partially unfold the first one so you can tuck in the last fold.
This section is incomplete!
Planned hexagon shaping instructions:
petal
sun
thorn
scissors
hemp leaf (asanoha)
others as I test out additional shapings
Deltahedra variations
Deltahedra variations
A deltahedron is a polyhedron consisting of only equilateral triangle faces. The delta-hex unit should be able to construct any deltahedra. So far, I have confirmed all 8 convex deltahedra, as well as the stellated octahedron.
Regular Tetrahedron (4 units)
A regular tetrahedron is a 4 sided pyramid and is the simplest construction possible.
A flat map of a tetrahedron.
View looking head-on at a tip of the pyramid, showing 3 of the 4 faces. The 4th face is hidden below.
View looking head-on at a face, showing 1 of the 4 faces. This obstructs the view of the 3 other faces.
Triangular bipyramid (6 units)
A triangular bipyramid attaches 2 3-unit pyramids at their bases.
A flat map of a triangular bipyramid.
A side view of the deltahedron showing 4 of the 6 faces.
View looking head-on at a pyramid tip, showing 3 faces.
Regular Octahedron (8 units)
A regular octahedron attaches 2 4-unit pyramids at their bases.
A flat map of a regular octahedron.
A side view of the deltahedron focusing on one of the faces. 3 other faces can be partially viewed.
View looking head-on at a pyramid tip, showing 4 faces.
Pentagonal bipyramid (10 units)
A pentagonal bipyramid attaches 2 5-unit pyramids at their bases.
A flat map of a pentagonal bipyramid.
A side view of the deltahedron with 4 faces visible. It's also easy to see in this view that the pyramids have become very short, but with very wide bases.
View looking head-on at a pyramid tip, showing 5 faces.
Snub disphenoid (12 units)
A snub disphenoid is a weird polyhedron that can be described 2 triangular 4-face bipyramids connected by 4 additional triangles (or described by several other unhelpful shape combinations.)
A flat map of a snub disphenoid.
One view of the deltahedron shows 5 faces connecting together, but not at consistent angles.
A different view of the deltahedron shows 4 faces connected together.
Triaugmented triangular prism (14 units)
A triaugmeted triangular prism attaches 3 3-unit pyramids with an extra triangle between the bases.
A flat map of a triaugmented triangular prism.
A side view of the deltahedron shows the pyramids at the top center, bottom left, and bottom right, connected by an additional face.
Another view of the deltahedron shows places where 5 faces connect.
Gyroelongated square prism (16 units)
A gyroelongated square prism attaches 2 4-unit pyramids with a square antiprism between.
A flat map of a gyroelongated square prism.
Viewing a pyramid head-on showing its four faces.
Viewing the deltahedron from the side shows the pyramids at both ends with a layer of faces between.
Regular icosahedron (20 units)
A regular icosahedron is recognizable to many people as a D20 dice. It attaches 2 5-unit pyramids with a pentagonal antiprism between.
A view centered on a tip, showing 5 faces connected to each other.
Viewing the deltahedron from the side shows the pyramids at both ends with a layer of faces between.
Stella octangula/ Stellated octahedron (24 units)
A stellated octahedron takes a regular octahedron and attaches pyramids to all of its faces. It creates a pleasantly spiky modular.
To create this modular, assemble 8 3-unit pyramids, then attach each pyramid to 3 others.
A view centered on a pyramid's tip, showing how its base connects to 3 other pyramids.
A view centered on pyramid bases, showing how 4 connect to each other.
Background
Background
The "delta-hex" modular unit is named because it is a triangular unit centered around a hexagon twist. Initially, I was working through "The Hex-agon" tessellation (a dense double flagstone pattern consisting of closed hexagon twists on one side and closed triangle twists on the other) and testing out fun variations I could do with a closed hexagon twist. I cut out a few hexagons and folded some fast 8-triangle grids with a central hexagon twist and began some variants, like Shuzo Fujimoto's windmill and others. I was noodling around with the edges of the star, folded a few tips in and made a triangle, and was reminded of Evan Zodl's Star Icosahedron modular. I decided to see if I could make my own modular unit and ended up with a central flap/pocket compared to Evan's corner flap/pocket setup. From there, I tested various deltrahedra (polyhedra whose faces are only composed of equilateral triangles) and was able to successfully create all the convex polyhedra as well as a personal favorite, the stellated octahedron (stella octangula).
The first units were folded in August 2025, with instructions shared shortly after.
This is the first modular that I designed myself!
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