An Origami Fractal

Wojtek BurczykWoj.burczyk@gmail.com

www.origami.edu.pl

3es Jornades Aplec De Didàctica Del Plegat

Per A Educadors

Badalona, 20-22 October 2017

Origami is a play for children, isn’t it?

Friedrich Froebel was very successful with his concept of

kindergarten. All of us have attended kindergarten and we

have folded simple Froebel style origami in our

kindergarten. So general public often perceives origami as

• a stupid play for small children;

• a didactic tool for small children (better case);

• a didactic tool for teaching of elementary geometry (case

of advanced person).

I will try to present origami as a tool for introducing modern

math concept.

© Wojtek Burczyk, 2017 2

Fractal

A fractal is an object or quantity that displays self-similarity,

in a somewhat technical sense, on all scales. The object

need not exhibit exactly the same structure at all scales,

but the same "type" of structures must appear on all scales.

(MathWorld).

Usually fractals are generated by computers as nice

graphic but it is possible to make an origami fractal.

© Wojtek Burczyk, 2017 3

Self similarity od A4

A4 paper is widely available (in Europe) and it shows nice

property of self similarity – half of a sheet is similar (the

same proportion) to the original sheet

© Wojtek Burczyk, 2017 4

Lets fold – a Bow-tie module

Take a waterbomb base, fold halfway of its height and squash – you will get simplest bow-tie.

Place a bow-tie at the end of a a strip.

© Wojtek Burczyk, 2017 5

Bow-tie – a brief history

Bow-tie may be traced back to Fröbelworks in 19th century. Paper of EleonoreHeewart Course in paper Folding – One of Froebel’s Occupations for Childrenoriginally published in 1895 present models we believe are examples of bow-ties: a fish and a candy.

© Wojtek Burczyk, 2017 6

Our developments

• Bow-tie offers excellent locking mechanism.

• 5 different locking mechanisms.

• Variety of origami models based on bow-ties:

- Easy to fold and accessible for unexperienced students (starting from kindergarten).

- Interesting from mathematical and aesthetical point of view.

© Wojtek Burczyk, 2017 7

Locking – easy and powerful

• We have discovered

a locking mechanism

– any flap ended with

a right angle fits to the

pocket in the triangle

at back side of a fish

module. And it is a

very firm lock.

© Wojtek Burczyk, 2017 8

Fish module

• Pocket

• Two flaps

• Fork and merge

mechanism

© Wojtek Burczyk, 2017

flap

pocket

9

Bow-tie motif – Pythagorean tree

As opposed to most of modular, bow-tie allows different sizes of modules that join together.

Flap may be of any size as long it has a right angle. Together with A4 format self-similarity leads to nice fractal forms

© Wojtek Burczyk, 2017 10

A Problem

Imagine that you continue process of

growing the tree and add smaller and

smaller branches.

What will become a problem:

• size of your table

or

• strength of your table?

© Wojtek Burczyk, 2017 11

Pitagorean tree

© Wojtek Burczyk, 2017 12

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Pythagorean tree artistic project

12 trees for the 12th Polish origami convention (Outdoor Origami Meeting 2013)

© Wojtek Burczyk, 2017

Bow-tie motif – Pythagorean tree

Random size of modules

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Conclusion

• Origami is a play for children, isn’t it?

• Actually it is not.

• There is some place for origami also at the

higher level of education.

• And there is lot of space for origami in fine

art domain (but that’s another story)

© Wojtek Burczyk, 2017 15

Thank you

Wojtek Burczyk

References

1. Mathworld - Weisstein, Eric W. MathWorld - A

Wolfram Web Resource.

http://mathworld.wolfram.com

2. Krystyna Burczyk. Kokardkowe koronki. Self-

published. Zabierzów, 2004

3. Eleonore Heewart Course in paper Folding –

One of Froebel’s Occupations for Children,

1895, in John Smith (ed) COET’91

Proceedingd of the First International

Conference on Origami in Education and

Therapy, British Origami Society 1992

© Wojtek Burczyk, 2017 17