Mathematical Objects


Models of this type are also automatically listed in: abstract, geometric
More restrictive types: balls and polyhedra, cubes and cuboids, fractal, intersecting planes, modular balls and polyhedra, modular cubes and cuboids, modular fractal, modular intersecting planes, other modular mathematical object, other modular polyhedron, modular spiky balls, modular woven polyhedron, other mathematical object (non-modular), other mathematical object, other polyhedra, recursive non-tileable model, spiky balls, recursive and periodic tessellations, recursive tessellation, woven polyhedron

Models representing a wide array of simple mathematical objects.

This page lists models of a single type. You might be interested in folding instructions instead.
Möbius Strip V (CBU)

Möbius Strip V (CBU)

A Möbius strip consisting of a single Conveyor Belt Unit (CBU) twisted and locked with itself by both ends. This is the cleanest representation of a Möbius s...

Hydrangea (Tant)

Hydrangea (Tant)

Another fold of Shuzo Fujimoto’s Hydrangea. I’ve kept two outermost leaf layers free of unnecessary creases. For deeper layers, it becomes much harder.

Trefoil Knot

Trefoil Knot

A trefoil knot from a single Cross Lap Unit.

Cube (BBU E7)

Cube (BBU E7)

Another cube from BBU-s: 6 × E7, 6 × D4 6 × A1.

Chrysanthemum

Chrysanthemum

This design is my modification of Shuzo Fujimoto’s Hydrangea. Despite the change being very minor, it allows the model to be shaped in a more three-dimension...

Lucky Star Fractals (10 different colors)

Lucky Star Fractals (10 different colors)

Ten level-3 Lucky Star Fractals, folded from metallic paper in different colors. This is just a small subset of how many times I have folded this model over ...

Hydrangea

Hydrangea

This is, along with Clover Folding, one of the oldest pictures of a tessellation folded by me (taken in June 2015).

Clover Folding

Clover Folding

Along with the Hydrangea, this is one of the oldest picture of a tessellation folded by me (taken in June 2015).

Clover Folding (blue)

Clover Folding (blue)

Along with the other Clover Folding model, this is the oldest picture of a tessellation folded by me (taken in June 2015).

90-Edge Buckyball (PHiZZ Variant IV)

90-Edge Buckyball (PHiZZ Variant IV)

This was an experiment with yet another PHiZZ variation of mine, conducted a few years ago. I chose too soft paper (or too large sheets) for this model which...

Hydrangea (Prägo Kristall paper)

Hydrangea (Prägo Kristall paper)

This is just a simple Hydrangea, designed by Shuzo Fujimoto, but I think it looks really nice in back light. Folded from Grünperga Kristall Prägo, a kind of ...

Two-Unit Cube II

Two-Unit Cube II

Another simple model in which a cube is built from just two units. See also: Two-Unit Cube I.

Hydrangea from book wrapping foil

Hydrangea from book wrapping foil

Model folded from transparent book wrapping fold which shows the internal structure of the model in an interesting way. Have a look at the detailed review of...

Spiked Icosahedron (Paper Airplane Sonobe)

Spiked Icosahedron (Paper Airplane Sonobe)

Spiked Icosahedron made from my new Sonobe variant, Paper Airplane Sonobe. When you look at an individual unit before assembling the model, you can spot a re...

Two-Unit Cube I

Two-Unit Cube I

This is a very simple modular origami design I recently came up with when revisiting my Oxi unit from a few years ago. The unit has folded edges on one side ...

Triangle Recursion

Triangle Recursion

This is the simplest recursive/fractal model I have come up with so far. It is folded from a square. Due to the very high shrinkage factor, which is almost 4...

Twisted Bird Base Fractal

Twisted Bird Base Fractal

This is a recursion of my Twisted Bird Base Tessellation. The recursive / fractal version looks interesting, but it’s rather hard to collapse.

Lucky Star Fractal Tessellation

Lucky Star Fractal Tessellation

This is a tessellation of the Lucky Star Fractal. The standalone star was designed independently by several people, starting with Shuzo Fujimoto. I don’t kno...

Lucky Star Fractal (Open Back)

Lucky Star Fractal (Open Back)

This model is a variant of the Lucky Star Fractal (aka Logarithmic Star), designed by myself and independently by many others before me, starting with Shuzo ...

Lucky Star Fractal (level 3)

Lucky Star Fractal (level 3)

A level-3 fold of Lucky Star Fractal, this time from gray metallic paper. I rarely fold a model multiple times, but this particular model is so nice I have f...

Lucky Star Fractal

Lucky Star Fractal

This is a recursive version of the Lucky Star molecule. Just like the non-recursive version, it can be tessellated or used for decorating a box. The back of ...

Rhombus Fractal

Rhombus Fractal

Rhombus Fractal, an origami tessellation designed by Endre Somos (and independently also by Joel Cooper and by Wei Fu), folded by me. You may also find inter...

Stacked Propellers Tessellation

Stacked Propellers Tessellation

A refold of my older design, Stacked Propellers Tessellation, this time as a complete tessellation rather than a single molecule.

Phantasma Fractal Variant

Phantasma Fractal Variant

This is my variant of Edward Mistretta’s model, Phantasma Fractal. It’s a nice design which can be folded as a single molecule, or tessellated.

Hamiltonian Cycle of Cube

Hamiltonian Cycle of Cube

My design for a single-sheet Hamiltonian cycle of a cube. Origami folded from a single long (just above 5:1) rectangle of paper. The bent frame is in typical...

Clover Folding Lookalike

Clover Folding Lookalike

This is my design which at first glance looks very similar to Shuzo Fujimoto’s Clover Folding but has a completely different crease pattern and design.

Clover Folding (22×22 grid)

Clover Folding (22×22 grid)

I hadn’t folded Shuzo Fujimoto’s Clover Folding for at least two years when I decided to fold one again after seeing some examples beautifully folded by John...

Hydrangea Cube (Harmony paper)

Hydrangea Cube (Harmony paper)

Shuzo Fujimoto’s Hydrangea can be used as a modular unit. The method was first published by Meenakshi Mukerji and then reinvented independently by myself. I ...

Hydrangea Heart

Hydrangea Heart

My modification of Shuzo Fujimoto’s Hydrangea into a heart shape. I found on flickr a similar model, designed by Francis Ow.

Lelum Cube

Lelum Cube

A modified version of Lelum Polelum Cube where one out of each pair of flaps is hidden.

Lelum Polelum Cube

Lelum Polelum Cube

A Cube from a unit I recently designed and later learned that was earlier designed independently by Saburo Kase. More details in the unit’s description.

Hydrangea with Leaves

Hydrangea with Leaves

A color-change variant of Shuzo Fujimoto’s Hydrangea, designed by Meenakshi Mukerji. Test fold based on unpublished diagrams, kindly provided by Meenakshi [u...

Stacked Hearts

Stacked Hearts

Stacked molecules of my Hearts Tessellation. This model is a fractal, and you can go on stacking heart upon heart indefinitely, each twice as large as the pr...

Leafless Hydrangea

Leafless Hydrangea

This Leafless Hydrangea model is a simple modification of Shuzo Fujimoto’s Hydrangea. It’s interesting how a simple change can modify a model’s appearance. J...

Mesos Logo (Cube-Hexagon Illusion)

Mesos Logo (Cube-Hexagon Illusion)

This is the logo of Apache Mesos (cluster management software) rendered in origami. A colleague at work suggested I try designing this object in origami afte...

Aton Kusudama

Aton Kusudama

Aton Kusudama, variant A (many more are possible). Spiked icosahedron made from 30 units. Designed and folded by me. More variants are possible, some with in...

Purple 90-Edge Buckyball (PHiZZ Variant II)

Purple 90-Edge Buckyball (PHiZZ Variant II)

90-edge buckyball made from a variation of Tom Hull’s PHiZZ unit. I know that other people have also designed this simple variant of the unit independently f...

Möbius Strip IV (CLU)

Möbius Strip IV (CLU)

The Möbius band (aka Möbius strip) is an interesting mathematical object, a single-sided surface. This origami version is folded using my Cross Lap Unit (CLU...

Lotus Cube

Lotus Cube

Lotus Cube, made from a variant of my BBU (Building Block Units). Even though it is possible to make a cube from just 6 lotus BBU units, such an assembly is ...

Six Intersecting Pentagrams (SIP)

Six Intersecting Pentagrams (SIP)

This modular woven polyhedron not only looks nice, but it also has a very interesting history of different people designing it independently. Those credited ...

Coaster Cube

Coaster Cube

This cube is made from a slightly modified variant of my Woven Slit Module (WSM). 36 units are used (6×4 = 24 for the faces and 12 for the edges), made from ...

Flower Icosahedron (StEM)

Flower Icosahedron (StEM)

This is an icosahedron (or dodecahedron, depending on how you look at it) made from a modified version of Sturdy Edge Module (StEM), a 90-degree unit variant...

Cube from Recursive Four-Sink Base

Cube from Recursive Four-Sink Base

This cube is made from six units, each of which is a recursive four-sink base modified for use as a module.

Fractal Pinwheel Cube

Fractal Pinwheel Cube

This is an example of using my Fractal Pinwheel as a modular unit. Due to small size, there’s only one level so the fractalness is not so clearly visible.

Fractal Pinwheel

Fractal Pinwheel

This pinwheel displays a fractal pattern with smaller pinwheels embedded inside larger ones, a series which can be continued indefinitely with thin enough pa...

Fenced Tiling of Clover Folding

Fenced Tiling of Clover Folding

I designed this tiling of Shuzo Fujimoto’s Clover Folding after I saw the tiling by Peter Budai and thought it would be better to make the borders between mo...

The Tower of Babel

The Tower of Babel

This rendition of the Tower of Babel consists of a series of square platforms placed one on top of another and rotated by 45 degrees at each level. This frac...

Down the Rabbit Hole

Down the Rabbit Hole

This tessellation consists of concentric square twists of growing size. The medium is self-adhesive holographic foil glued onto tracing paper. The spiral is ...

Spiked Octahedron (WSM)

Spiked Octahedron (WSM)

Spiked octahedron made from my Woven Slit Module (WSM), 48 units from 3:1 paper.

Hydrangea Picture Frame

Hydrangea Picture Frame

This picture frame can hold a standard 15×10 cm photograph. It consists of four molecules of the Hydrangea Tessellation (designed by Shuzo Fujimoto), spaced ...

Hydrangea Icosahedron

Hydrangea Icosahedron

After I made a Hydrangea Cube, Hydrangea Icosahedron was the next logical step. Just as in the cube, the Hydrangea Tessellation by Shuzo Fujimoto is used as ...

Cube (2:1 paper, slits outside)

Cube (2:1 paper, slits outside)

In this assembly method, each of the cube’s faces is made of two modules which are both attached to both perpendicular modules in the same way. Together with...

Adjustable Cube

Adjustable Cube

This cube is a mechanical toy. Its size can be adjusted: the cube can grow or shrink by a factor of about two. It starts out as a cube with a pattern resembl...

Knobby Cube

Knobby Cube

Another combination of Building Block Units and tessellations, this time Fujimoto’s Clover Folding, folded without the decorative margin. 18 modules: 6 × BB...

Hydrangea Cube

Hydrangea Cube

I came up with the idea of connecting Hydrangeas to form a modular origami design independently, then found out Meenakshi Mukerji had published it in her boo...

Square Weave Cube

Square Weave Cube

This is a modular cube made of six Square Weave Tessellations. The connection method is mine, the authorship of the Square Weave Tessellation seems to be dis...

Clover Cube

Clover Cube

This model is a combination of Building Block Units and Fujimoto’s Clover Folding. The models amounts to 18 units, 12 of which are BBUs (6 × D10 variant, 6 ×...

Möbius Strip III (BBU)

Möbius Strip III (BBU)

A single-sided surface, the Möbius Band is one of the more interesting mathematical objects that can be reproduced in origami.

Annapurna from StEM modules

Annapurna from StEM modules

Annapurna (also known as ten intersecting triangles or 10 × 3 × 1 polypolyhedron) was designed by Robert Lang, but the model presented here uses my Sturdy Ed...

Makalu

Makalu

Makalu is one of the models in Robert Lang’s Himalayan Peaks series. Its more scientific name is six intersecting pentagons, or: 6 × 5 × 1 polypolyhedron. Se...

Gasherbrum

Gasherbrum

This is the simplest of Robert J. Lang’s polypolyhedra. A more descriptive name of this model is four intersecting triangles, or 4 × 3 × 1 polypolyhedron.

Spiked Icosahedron (BBU D2)

Spiked Icosahedron (BBU D2)

The structure of this model is similar to spiked icosahedra made with variants of the Sonobe unit and other similar modules. However, in the case of BBU, a t...

Rectangular Cuboid

Rectangular Cuboid

This model demonstrates how Building Block Units can be modified to form rectangular rather than square faces. Just like the cube, this model uses 12 modules...

Hamiltonia Cycle of the Cube

Hamiltonia Cycle of the Cube

A Hamiltonian cycle is a closed path on a polyhedron which visits each vertex exactly once. This model represents such a path for a cube. It can also be used...

Expanded Hexagonal Prism

Expanded Hexagonal Prism

This is a shape created by placing cubes on the outer square walls of a hexagonal prism. This way, the outer outline becomes a dodecagonal prism. Seen from t...

Cube (BBU E10)

Cube (BBU E10)

Due to the E10 tile’s small flaps, it can’t be directly attached to the flaps of inner A1 tiles. An additional “sizing” layer of A2 tiles is needed for prope...

Cube (BBU D9)

Cube (BBU D9)

Cube from 12 modules: 6 × D9, 6 × A1.

14-Spoked Wheel

14-Spoked Wheel

Mathematically speaking, this wheel is a tetradecagonal prism. This construction, which uses a mix of units made from 1:√2 and 1:2√2 paper, isn’t mathematica...

Large Cube

Large Cube

This cube uses PVM Edge Connector Units to create extra distance between the Vertex Modules.

Cube from Sunken Vertex Units

Cube from Sunken Vertex Units

The result of using the sunken variant of PVM Vertex Unit is a cube with four vertices replaced by inverted pyramids.

Toshie’s Jewel (StEM)

Toshie’s Jewel (StEM)

Normally, Toshie’s jewel is made from Sonobe units, but this one is made from StEM units instead.

Spiked Icosahedron (StEM)

Spiked Icosahedron (StEM)

In this model, each face of an icosahedron was replaced with a triangular pyramid made from three units.

Spiked Icosahedron (StEM face variant)

Spiked Icosahedron (StEM face variant)

The unit is a variant of an edge unit; I call usage like this the “face variant” since the unit covers a face rather than an edge of the solid. When I invent...

Octahedron with Inverted Spikes on all Faces

Octahedron with Inverted Spikes on all Faces

This model’s structure is an octahedron whose each face was replaced with a pyramid of three equilateral right triangles, pointing inwards. Units are located...

Menger Sponge, level 1½ (StEM)

Menger Sponge, level 1½ (StEM)

The modules’ shape makes this level 1 model look even closer to a level 2 model than the Penultimate Module version. The hole in each small square is exactly...

Tetrahedron (StEM)

Tetrahedron (StEM)

This model (first from the left) is compared here with some other simple polyhedra folded from the same kind of module. Note how the tetrahedron looks almost...

Tetrahedron (SEU Sonobe)

Tetrahedron (SEU Sonobe)

This model (first in bottom row) is shown compared to other models folded from SEU units made from 2:1 and square paper (top and bottom row, respectively). N...

Tetrahedron (SEU from 2:1 paper)

Tetrahedron (SEU from 2:1 paper)

This model (first in top row) is shown compared to other models folded from SEU units made from 2:1 and square paper (top and bottom row, respectively). Note...

Spiked Octahedron (rotated link method)

Spiked Octahedron (rotated link method)

This model demonstrates the rotated link connection method that can be applied to SEU units folded from square paper, which can be considered a Sonobe varian...

Spiked Icosahedron (Sonobe link method)

Spiked Icosahedron (Sonobe link method)

This model demonstrates the Sonobe link connection method that can be applied to SEU units folded from square paper, which can be considered a Sonobe variant...

Spiked Icosahedron (SEU link method)

Spiked Icosahedron (SEU link method)

This model demonstrates the SEU link connection method that can be applied to SEU units folded from square paper, which can be considered a Sonobe variant. T...

Ring

Ring

This ring can also be worn as a headband. It uses a non-standard way of connecting the modules. Any even number of modules can be connected this way, though ...

Octahedron (StEM, modules pointing outside)

Octahedron (StEM, modules pointing outside)

This model (first from the right, top row) is compared here with some other simple polyhedra folded from the same kind of module. The two octahedra demonstra...

Octahedron (StEM, modules pointing inside)

Octahedron (StEM, modules pointing inside)

This model (first from the right, bottom row) is compared here with some other simple polyhedra folded from the same kind of module. The two octahedra demons...

Octahedron (SEU Sonobe)

Octahedron (SEU Sonobe)

This model (last in bottom row) is shown compared to other models folded from SEU units made from 2:1 and square paper (top and bottom row, respectively). No...

Octahedron (SEU from 2:1 paper)

Octahedron (SEU from 2:1 paper)

This model (last in top row) is shown compared to other models folded from SEU units made from 2:1 and square paper (top and bottom row, respectively). Note ...

Cube (StEM)

Cube (StEM)

This model (second from the left) is compared here with some other simple polyhedra folded from the same kind of module.

Cube (SEU Sonobe)

Cube (SEU Sonobe)

This model (second in bottom row) is shown compared to other models folded from SEU units made from 2:1 and square paper (top and bottom row, respectively).

Cube (SEU from 2:1 paper)

Cube (SEU from 2:1 paper)

This model (second in top row) is shown compared to other models folded from SEU units made from 2:1 and square paper (top and bottom row, respectively).

Menger Sponge (level 2)

Menger Sponge (level 2)

This is a labor-intensive, but very satisfying to fold model. Some people have managed to go as far as level 3 but even level 2 was quite challenging. About ...

Spiked Icosahedron

Spiked Icosahedron

I designed the simple unit used for this model and later learned that it had been already published before by Jose Arley Moreno.

Ticket Menger Sponge

Ticket Menger Sponge

Model folded from Warsaw public transport tickets (back side with magnetic strip visible). 192 modules: 120 for the body and 72 for coating.

Ticket Cube

Ticket Cube

I folded this business card cube from Warsaw public transport tickets rather than from business cards. 12 modules: 6 for the body and 6 for the coating.

Truncated Octahedron

Truncated Octahedron

This was one of my early modifications of the 60° unit. Note that in this modification, the angle at the module’s tip is NOT 60 degrees.

Flower Icosahedron (60°)

Flower Icosahedron (60°)

Compare with a dodecahedron constructed from units modified by me in a similar manner, and with a model with the same structure but using StEM units.

Spiked Icosahedron (Sonobe variant)

Spiked Icosahedron (Sonobe variant)

I think this is my first Sonobe variant. Since it’s one of the simplest modifications possible, it has probably been independently discovered by many others.

Icosahedron

Icosahedron

You can compare this model, which uses straight, unmodified units, with two models made from the same units after slight modification: Flower Icosahedron and...

Umbrella Dodecahedron

Umbrella Dodecahedron

The module, originally designed just for folding this dodecahedron, can be also used for other kinds of models. See, for example, this spiked icosahedron.

Poinsettia Ball

Poinsettia Ball

Model is placed near a real Poinsettia flower for comparison.

Decorated Dodecahedron (Penultimate unit)

Decorated Dodecahedron (Penultimate unit)

This model is made from 90 modules (modified variant for triangular faces). Each face of the dodecahedron is made from a 5-triangle group, where the triangul...

FIT (Five Intersecting Tetrahedra)

FIT (Five Intersecting Tetrahedra)

At only 30 modules, this model is still much more challenging than most models with several times that many units, but also a lot of fun to fold. See the lin...

Truncated Cube (PHiZZ)

Truncated Cube (PHiZZ)

Generally, PHiZZ units are always connected in such way that three modules meet at each vertex. However, one can connect just two modules at some points, thu...

Modified Buckyball (120 edges)

Modified Buckyball (120 edges)

This is my experiment in modular origami made from two different types of units: 60 PHiZZ and 60 Penultimate units. These two kinds of modules are quite simi...

Jitterbug Icosidodecahedron

Jitterbug Icosidodecahedron

You can squeeze this model and transform it into an icosahedron, closing the empty space between units. This is called the jitterbug transformation.

Decorated Icosidodecahedron

Decorated Icosidodecahedron

One of the larger models I have designed, this icosidodecahedron has pentagonal faces made up of small triangular pyramids and triangular faces replaced with...

Klein Bottle

Klein Bottle

This model was quite difficult to design, as the two sides of surfaces made with PHiZZ modules differ a lot (due to the presence of “bumps” where units join)...

Spiked Pentakisdodecahedron (Sonobe)

Spiked Pentakisdodecahedron (Sonobe)

There is one spike placed over two adjacent faces of the pentakisdodecahedron in this model. I haven’t checked if the angles actually add up, so it might be ...

WXYZ

WXYZ

Model is also known as WXYZ Diamonds.

Steinhaus Puzzle

Steinhaus Puzzle

This puzzle, described in Hugo Steinhaus’ book Kalejdoskop matematyczny (Mathematical Snapshots, literally Mathematical Kaleidoscope) consists of six pieces,...

Spiked Icosahedron (Trimodule)

Spiked Icosahedron (Trimodule)

Apart from this basic version, I also made a variant of this model which has additional “fins” on the icosahedron’s edges.

Spiked Icosahedron (Penultimate Unit)

Spiked Icosahedron (Penultimate Unit)

One way of looking at this model is to see it as an icosahedron with a pyramid placed on each triangular face. Another is seeing it as a dodecahedron where e...

Möbius Strip II (Trimodule)

Möbius Strip II (Trimodule)

This is one of the rather few modular origami designs which use an odd number of units. Compare also with another similar model.

Möbius Strip I (Trimodule)

Möbius Strip I (Trimodule)

This is one of the rather few modular origami designs which require an odd number of modules. Compare also with another similar model.

Menger Sponge, level 1½ (Penultimate unit)

Menger Sponge, level 1½ (Penultimate unit)

Thanks to the modules’ shape and the holes created in the spaces between them, this model looks almost like a level 2 Menger sponge even though it is actuall...

Large Icosahedron

Large Icosahedron

This icosahedron has nine triangular pyramids pointing inwards on each face. The same shape can also be described as a truncated icosahedron whose each face ...

Koch Snowflake, level 1 (Trimodule)

Koch Snowflake, level 1 (Trimodule)

A level-1 Koch snowflake is just a simple hexagonal star, and this is the way of connecting the Trimodule units originally suggested in Nick Robinson’s instr...

3D Koch Snowflake (Trimodule)

3D Koch Snowflake (Trimodule)

This fractal is an analogue of the standard Koch snowflake. Level 0 is a tetrahedron. In each iteration, a tetrahedron with half the edge length is placed in...

Blintz Icosidodecahedron

Blintz Icosidodecahedron

There are six intersecting planar surfaces, each in the shape of pentagonal star, in this model. This leads to the most popular coloring with six different c...