3D Printing is about being able to print any object layer by layer.

I was cogitating on this one day and thought to myself:

## Are there any 3 dimensional items we can’t print layer by layer?

After all, it is assumed that a 3D printer can print ANY object – but is this proved? Is there a Mathematical Theorem that proves we can 3D Print anything?

I know this is very much an **“It’s obvious”** question, but sometimes the devil is in the details. And if there are items we can’t print – we need to know about them now. In short – this stuff is **IMPORTANT**.

It just so happened that weekend I was going to a gathering where Associate Professer Alan Branford, Director of Studies in Mathematics and Statistics at Flinders University in South Australia, was attending.

“Prof” as he is nicknamed is a tall well spoken man with an enormous passion for mathematics.

## Mathematics to the Rescue

I gathered my nerves and walked up to him:

“Hi Prof”

“Hello Andy – how are you doing?”

“I’m doing good. Prof, I have a mathematics questions”

The Prof spat out his Pinot Noir, looked at me through gunfighter eyes. He straightened up, but now his trigger finger was twitching, and he carefully remarked “Proceed”.

Nervously I asked “Prof, I run a blog on 3D Printing.”

“Fubini’s Theorem” Prof interjected

“.. and I was wondering if there was a proof that …”

“Fubini’s Theorem” he said

“.. that we are able to express any 3D Printed object as layers of 2D planes”

“Yes, Fubini’s Theorem”

So there you go. Fubini’s Theorem is the one we care about.

I have reproduced the guts of it here:

I am not a mathematical person once I need to go beyond algebra. So I asked Prof what it meant in blog author language.

He said that Fubini’s Theorem states that an object of ** n** dimensions can be represented as a spectrum of layers of shapes of

**layers.**

*(n-1)-dimensional*This means that a 3 dimensional shape (any shape in the real world) can be represented as layers of 2 dimensional shapes.

## But…

“Well that is exactly what I wanted – thanks Prof!” I turned to walk away

“WAIT!”Prof screamed in righteous fury“There are some cases you need to know about!”

After a long discussion it turns out that Mathematicians are a fairly pathological bunch. Always coming up with strange cases that prove or disprove theories. They think it is funny.

Prof sums it up in this quote:

It turns out that Fubini’s Theorem holds for all what could be called realistic physical objects, but Mathematicians can dream up bizarre things where it doesn’t.

You usually need a postgraduate Math degree to even understand their description – they are certainly not something of the real world – but to be a pedant we must concede these creatures exist in theory.

A more practical limitation is the slicing resolution and also the issue of physical stability during the layering – these are essentially engineering issues.

What this means is that **when we are printing, the 2D slices are pretty much always an approximation of the real shape. **

## Does this approximation matter?

No, not really, in most cases. Generally the precision of printers is limited by the material we are using – for example the Reprap Prusha Mendel 3D Printer currently prints to 0.3mm resolution – so when we are calculating the slices, we just have it in mind that the printer can’t print anything smaller than a 0.3mm spot.

The software we use for slicing does take this approximation into account so we are safe there.

## So now you know

Fubini’s Theorem is your friend – so if anyone ever asks you if you can print ANYTHING on a 3D Printer – say “Yes, any real life physical object”.

Only if they get all Mathematician on you, can you state “Fubini’s Theorem, I know of it Horatio!”

#### A big thanks to Prof

Thanks Alan for being so forthcoming with this information! I battled through mathematician forums, manifold mathematics until I was a sobbing wreck! Thanks mate.

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Nice post!

One minor correction, though: https://plus.google.com/u/0/110900759695819888399/posts/V94YZrNEkSN

Thanks for that Dave – I did actually know about the 0.1 resolution. It is bloody exciting!

Could you print a spherical mirror for use in a telescope? I think the resolution of the current devices isn’t suitable, so the resolution _does_ matter.

You could print the mirror – but you would still need to polish it using conventional methods.

Old post, but still:

Any *shape* not “any real life object”

[…] question is that if there is anything that you cannot print. The short answer is NO. Thanks to Fubini’s Theorem. This entry was posted in Applications, Calculus/Analysis. Bookmark the permalink. ← […]

[…] what important mathematical principles apply to 3D Printing — and his professor said … Fubini’s Theorem! Here’s a selection of his post covering this […]