Embryo Morphogenesis is a mathematical mini-universe. It is governed by laws which your students must figure out. Mini-mathematical universes like this are¬†possibly a great way to learn the scientific method. It still needs work. If you’re embarking on this exercise you do so at your own risk ūüėČ Unlike the real world where scientific experiments are costly, lengthy and awkward, these scientific experiments are cheap, fast, and precise. Real world science is essential, but these mini-mathematical universes offer a quick fix.

Try to figure out what is going on here…

Embryo Morphogenesis differs from creature to creature. This one has five stages.

Try to figure out some of the laws that apply to this mini-mathematical universe.

The next slides will take you on a walkabout through the transitions of the first (yellow) creature that you saw on the previous slide.


We can walk around these embryos… moving from one to another by following the arrows. In the next slides I’ll take you on one such walk going clockwise from the Toothless Granny (centre top.) I’ve named the embryos just to make discussion easier.

Here are patterns sheets including the one pictured here. Do NOT hand them all out at once. Present them as in this slide show.


Here is “Toothless Granny.”


Here is “Surprised Spartan.”

It looks a little bit like the “Toothless Granny.”


How are they changing?


What is the underlying structure?

Now we loop back to the beginning…

This is only one path. We can go back and forth following the arrows any way we choose.

This is just a reminder of the overall pattern for this embryo.

Hints will be provided, so if you still don’t understand what is going on – don’t worry.¬†It’s a small difficulty compared to what scientists face in understanding¬†the real world.


Embryo morphogenesis of this dove is worthy of contemplation. Can you figure out what the next slide will look like?


If you got it right, you will have no problem completing the whole pattern.

If you did not get this right, then try to predict what the next slide will look like.


Did you get it?

Of course the names are ridiculous… but the pattern is not. There is a logical progression.

What does the next slide look like?


That was a trick question. There is no other shape. There are only three shapes here.

It is time for the first hint. The next three slides show some of the underlying structure of this last embryo.


Hint 1/3

Hint 2/3

Hint 3/3

There will be more hints later, but I hope that was insightful.


The next slide will show the pentagon over-top the image as you saw in the three hints.

Can you figure out what the pentagon looks like?


Okay – now try to figure out the¬†whole pattern for this Embryo. I’ll reveal them one at a time in a walk through…

There are two directions to go now… I’ll flip the bottom half circle.

Now I’m stuck. I guess I’ll have to go back to the stuffed cheeks…

Now I’m¬†going to flip the smallest half circle.


Let’s see the whole pattern for this embryo morphogenesis.


You can see for the first time we were not able to walk through – visiting every shape – without visiting a shape twice.

Stop now and try to figure out the pattern.

I’ll take you on a tour…

There is more than one way to travel through these embryos.

There is still more structure, but it is difficult to see. Let me give you a sample from each¬†embyo¬†in order and see if you can figure out even more structure…

The yellow one… followed by…

After seeing the five in the right order there will be a hint.

Yellow – Orange –

Yellow – Orange – Pink –

Yellow – Orange – Pink – Blue –

Yellow – Orange – Pink – Blue – Red

It is difficult to see a pattern, but look at the next five slides…

Yellow –

Yellow – Orange –

Yellow – Orange – Pink –

Yellow – Orange – Pink – Blue –

Yellow – Orange – Pink – Blue – Red

Still don’t have an idea of this last bit of structure?

Look at the final slide…


If you are stuck on any part of this mini-universe, do email me. This was a lot of work. If you want more mini-universes I’ll need to have encouragement and kudos ūüėČ

Here are patterns sheets for your students to explore. Do not hand them all out at once. Present them as in this slide show.

Embryo Morphogenesis

This activity is still in development

(MathPickle, 2015)

After students have struggled with the hard version of Embryo Morphogenesis above, their natural inclination as mathematicians should be to simplify the problem. They should ask “How would a simple shape¬†behave?” Mathematicians are always seeking to understand simple stuff in order to illuminate the more difficult stuff.


The simplest shape they could think of should be an equilateral triangle or a square. How does embryo morphogenesis work for these two shapes. They should first take a stab at it.

The embryo above is from a square. Find the full embryo morphogenesis pattern for the square and the equilateral triangle. The problem is still ill-defined, so there is flexibility. In the next slides we are going to look at a two solutions for the square.


The first solution makes all possible folds. We understand that two half circles cannot intersect each other from the first Embryo Morphogenesis slide show above. However, we have never before encountered the question of what happens when they intersect at exactly one point. The extreme left and right embryos are examples of this extreme.


Your students may have chosen to omit them or leave them in. Both are perfectly reasonable interpretations of how embryo morphogenesis works.

Just in case you have trouble visualizing the square template in each embryo…

The first embryo morphogenesis did not have multiple embryos that were identical except for rotation.

An alternate, and totally plausible, interpretation of embryo morphogenesis is to group all identical embryos equivalent.


Now do the same for an equilateral triangle…

Assuming that rotations are different…

Assuming that rotations are different…

…with the equilateral triangles superimposed…

Assuming that rotations of the same shape are NOT¬†different…


See if any of your students can solve the regular pentagon.


Here is a hint. You can download this puzzle-sheet here.


Here is the answer.


Now let’s simplify this pattern by clumping together all shapes that are the same (after being¬†rotated).


Do the same pattern for¬†a regular hexagon. Only try to find the simplified pattern found by clumping together similar shapes…


Download a pdf of this puzzle-sheet here. This is the same pdf that was offered earlier. It is the second page.

Hope you enjoyed exploring this mini universe ūüėČ


Embryo Morphogenesis II

(MathPickle, 2015)

Standards for Mathematical Practice

MathPickle puzzle and game designs engage a wide spectrum of student abilities while targeting the following Standards for Mathematical Practice:

MP1 Toughen up!

Students develop grit and resiliency in the face of nasty, thorny problems. It is the most sought after skill for our students.

MP2 Think abstractly!

Students take problems and reformat them mathematically. This is helpful because mathematics lets them use powerful operations like addition.

MP3 Work together!

Students discuss their strategies to collaboratively solve a problem and identify missteps in a failed solution. MathPickle recommends pairing up students for all its puzzles.

MP4 Model reality!

Students create a model that mimics the real world. Discoveries made by manipulating the model often hint at something in the real world.

MP5 Use the right tools!

Students should use the right tools: 0-99 wall charts, graph paper, mathigon.org. etc.

MP6 Be precise!

Students learn to communicate using precise terminology. MathPickle encourages students not only to use the precise terms of others, but to invent and rigorously define their own terms.

MP7 Be observant!

Students learn to identify patterns. This is one of the things that the human brain does very well. We sometimes even identify patterns that don't really exist ūüėČ

MP8 Be lazy!?!

Students learn to seek for shortcuts. Why would you want to add the numbers one through a hundred if you can find an easier way to do it?


Please use MathPickle in your classrooms. If you have improvements to make, please contact me. I'll give you credit and kudos ūüėČ For a free poster of MathPickle's ideas on elementary math education go here.

Gordon Hamilton

(MMath, PhD)