The Magic Mathematical Formula that Explains Why We Can 3D Print Any Physical Object

Posted by in What's Hot

In theory, of course there are material restrictions but any physical object can be described using Fubini’s Theorem and can therefore be 3D printed. My handsome Aussie friend over at 3D Printing is the Future consulted Professor Alan Branford of Flinders University to get the lowdown.

“He said that Fubini’s Theorem states that an object of n dimensions can be represented as a spectrum of layers of shapes of (n-1)-dimensional  layers.

This means that a 3 dimensional shape (any shape in the real world) can be represented as layers of 2 dimensional shapes.”

For the full extract of the conversation head over to 3D Printing is the Future to eavesdrop on the witty banter.

There are of course also mathematical objects that can ONLY be produced using 3D printing, or as we like to say here at Shapeways, making the impossible possible


Shapeways = (impossible – im) x 


  1. Aaron T.

    Thanks for clarifying this Duann!

    I’m going to print this out, so if I’m at a party or hanging out in a bar and someone hasn’t heard of 3D printing, I’ll start with the “layer-by-layer” gesture, and if that doesn’t work I’ll bust out Fubini’s Theorem.

    1. Duann

      yeah, nothing seduces the ladies (or lads) like the ‘F-Theorem’.

  2. Joe Slote

    If only we could print out functions of a couple variables and redefine them *after* printing…hmm…

    1. Duann


      Cornell and MIT are both looking at this concept,

      Self assembling materials that can reassemble themselves to adapt, repair, etc….

  3. George Bell

    I think you can also assume f(x,y)=0 or 1, unless your printer can somehow print different densities. No need making math more complicated than necessary!

    Printing an object with infinite volume would be difficult, to say the least. The cost would also be infinite! So I think we can safely assume the assumptions are satisfied.

  4. Aaron T.

    I know you’re joking, Duann, and so was I, but to be serious here, I think the fact that people try to make every social interaction about sex is the single worst part of our culture, and has made me lose almost all interest in meeting or speaking to anyone at all. I am sick of nonsense being projected onto me as a result of other people’s bad behavior. How can we change this aspect of our society?

    1. Duann

      Hey Aaron,
      Apparently Mahatma Gandhi said “We must be the change we want to see in the world”
      That and 3D print a wedding ring, that usually cools libido’s.

      Apologies for any offense.

  5. Aaron T.

    No offense taken Duann. I am just making my perspective clear, as our friend Gandhi would have it. This is especially important on the internet, where low and common stereotypes mascarade as worldly knowledge. Shapeways is usually a refreshing exception, which is why you see me here.

  6. Luis


    How would scaling be implemented?

    @Aaron: Inasmuch as your voice is valid, so is that of those you consider invalid. Otherwise others will see you as putting their opinion down and see you in that light, so you’re creating another issue.
    You then perpetuate it by provoking a response.
    Besides: Without it you wouldn’t be here.



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