Phizz Unit Origami: Torus

Phizz Unit - TorusThe Phizz unit torus happens to be one of the most stunning, yet challenging, modular origami pieces I have done yet. It has 360 total units and took me a few months to complete. Once complete though, the design itself is surprisingly rigid and is really fun to look at and hold.

There are a few reasons why it took so long to complete. First, the paper I used was old velum. While giving it a vintage look, the paper was also more fragile and a lot of pieces tore resulting in having to fold more units than necessary. Second, I had no diagram to complete this. I only had a small picture that I found while browsing the Web. I had mapped the design out in my head as well as on paper a few different times, each time realizing that my mapping was wrong. I finally figured it out after 5 different iterations. The third reason it took so long was due to frustration. Rebuilding it over and over forced me to stop and start over a bunch of times which really taxed my patience.

Even though it was the most difficult modular piece I have done yet, it still inspires me to build another one with different paper and a fun color design.

Phizz Unit Diagram

phizz unit
PDF phizz-unit.pdf

Torus Directions

Phizz torus wireframeTo complete this design, I have a few of suggestions that will help the process:

  • Use the torus diagram to the right as a map on how to build each shape and its surrounding shapes
  • The numbers in the diagram tell how many sides each shape has not how many to build
  • Build the next shape onto the one previously built
  • Keep connecting shapes until you build a ring
  • To build the other half, start replicating the wireframe on the inner edge of the first ring
  • Build shapes until it connects to the outer side, creating the first ring of the torus
  • Keep building more connecting shapes, the rest builds itself from there
  • The yellow hexagrams are the shared polygons, meaning there will be only one ring of each of these versus two
  • Be patient, it’s awesome when it’s done

Close Ups

Phizz Unit TorusPhizz Unit Torus

Phizz Unit TorusPhizz Unit Torus

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  1. Hossein

    just woooow
    you are the best
    thank you bro.

  2. Thea

    Hey there! By 360 units, do you mean 360 sets of the three parts, so 3 sheets per unit, or one sheet per unit?

    1. 360 individual sheets of paper folded into Phizz units.

  3. Evan Walters

    I’m going to make one of these out of post-it notes. What’s cool is you could probobly make one of these out of sonomb units.

  4. Edi

    Hello, Chaunce Dolan! Wonderful 🙂
    I made one toro here:

    I put your site as reference.


    1. Thanks. Love the colors, it really makes it that much more impressive!

  5. Anna

    How many heptagrams are there?

  6. What size squares are you using and what size ring does it yield? I want to do one for Christmas, so I want it to be large enough to hang on my front door.

    1. It has been many years since I did this. I want to say that I used 3″ square sheets, which yielded a torus diameter of approx. 16″.

  7. Chaunce,

    I’d like to know where to get old velum paper(please advise).

    I made torus from 81 units, from 105 units, 240 and 555 units.
    In this process I realized that the quality of the paper and the coordination of the colors are very important factors.
    The 555 unit torus is made of pentagons, hexagons and heptagons and it is made of seven shades of yellow (starting from “almost pumpkin” color inside to the most faded yellow on the outside – “almost vanilla” color).

    You may see several pictures of these tori in our website
    ( in our Modular Origami Gallery.
    Go to, thereafter choose “Modular Origami” and click on:”visit gallery” – you may choose one of six galleries: one of those is the “PHIZZ Gallery”

    here picture #9, #10 and #11 shows the smallest torus I know (that is made of 81 units)
    picture #12, #13, #14 shows the torus from 105 units, #15 shows the torus from 240 units and #16, #17, #18, #19 shows various pictures of the torus made of 555 units.
    If you look carefully, you may see, that I made an error in the coloring with the pumpkin color, or maybe with the next shade on one part of the surface.

    I wonder what color combinations have you tried?


  8. ayush j

    Hey thanks for the diagram, really breaks down a seemingly complex form to a simple method. I am half way through mine, will post images when I’m done.
    I used a note pad (the one with pre-cut sqaure note leaves)to make my phizz units, thought that was a smart thing to do! but the paper itself was so thick, it numbed my fingers folding it in place, But I thin k at the same time it lends a good structural strength to the piece.
    I have googled online a lot about structures made from these units, but not much variety turns up, I was just wondering if you are planning some more pieces like these? maybe more complex and awesome!
    Also, I am very curious so as to how did you figure out the polygonal diagram of the torus? Did you model it in 3D and split it into polygons or did you count the polygons on an existing photograph 😉 or just plain old trial and error? I am wondering if one could apply the same principal to make more complex forms

    1. I kind of figured the design out from inspecting a grainy photo of the design that someone else had done. It really was a trial and error thing though. I ended up building and rebuilding it something like 4 times before I nailed the design. I would really enjoy making new constructs with the Phizz unit.

      1. ayush j

        you can see the final torus here, I think its turned out pretty cool thanks for your flat pattern. The most difficult part of the whole assembly was the last bit, when the two part have to be joined to each other, being a closed solid, the piece gets very stiff hence a little difficult. Also since the paper starts giving away its structural strength on multiple folding-unfolding, joints start opening up, I actually had to replace some of them with fresh units. But its worth the effort, as you can see in the photographs, it looks quite cool as a lighting feature in my room 🙂

        1. Great work. The pictures look great, nice lighting too.

  9. Ralph

    It took me a while to figure out what was going on in the build but I think I understand. I’d like to offer an alternative way of doing the build.
    Begin by making a row of 11 hexagons. This should lie flat with the pointed parts of the joints pointing upwards. Lift the ends and join them, making a 12th hexagon where the ends of the row join. You now have a ring of 12 hexagons with the points of the joints pointing inward. This is the inner ring of the torus. (These are represented by the lower two shaded hexagons in the diagram above. The diagram represents one half of a one sixth wedge of the torus.)
    Laying the ring of hexagons on its side, there are 12 notches on the top of this ring where the hexagons join. Use each notch as a vertex of a heptagon and make a ring of 12 heptagons on top of your ring of hexagons. Flip the model over and make another ring of 12 heptagons on the other side. You now have a cylinder that flares outward at the top and bottom.
    In the notches between heptagons, make pentagons. You will have 12 pentagons on each side of the model but unlike the earlier hexagons and heptagons, the pentagons do not share sides with one another.
    Now the model has scalloped sides, where each indentation has three sides, the middle one is an edge of a heptagon and the outer ones are sides of a pentagon. Use these three sides as three sides of a new hexagon. So you are adding another ring of 12 hexagons on each side of the model but this time the hexagons do not touch one another. The sides of your model are now curving in toward one another.
    You can now join up the seam by adding two new row of hexagons in the scallops between the previous hexagons, but this time the hexagons you make on one side of the model will each share an edge with the hexagons on the other side of the model. As you join the two sides of the torus, the shaded hexagons at the top of the diagram above will spontaneously form. Those shaded hexagons at the top of the diagram represent the outermost ring of the torus. There will be 12 of these.
    In the diagram above there are 2 shaded hexagons at the bottom and 3 at the top. This is a bit confusing. The two at the bottom are not shared with the wedges to the left and right so if you make six copies of the diagram above there will be 12 hexagons in the inner ring. The shaded hexagons at the top of the diagram are shared, that is, the rightmost hexagon of this unit is the leftmost hexagon of the next unit so if you make six of these units joined together, the shaded hexagons at the top still represent a row of 12 hexagons.
    I hope this helps.

    1. nice job, that looks awesome