Extruding Towers by Serially Grafting FrustaCHENG HerngYi herngyi@mit.edu

Massachusetts Institute of Technology

mailto:herngyi@mit.edu

IntroductionAutomated origami design: demo

Extrusion Origami

Cheng (2010 – 5OSME)

Cheng (2012)

Ovadya (2009)

Mitani (2009)

Related Work

ORI-REVO,Jun Mitani

Origami Flanged Pots,Robert Lang

Related Work

Mitani Lang Cheng

“flanges”

Grafting

i. Fold an origami (e.g. pattern of scales)

ii. Treat the product as a fresh sheet of paper

iii. Fold another origami (e.g. fish) to combine

Image source: Lang, Koi, opus 425, The Math and Magic of Origami, TED.com (2008)

Grafting

i. Fold an origami

ii. Treat the product as a fresh sheet of paper

iii. Fold another origami to combine

Modularized Design

Break solid into smaller frusta

Frustum(pl. frusta)

Right Frustum

Positive Frustum CP

Special case of biplanar!

Negative Frustum CP

Central Body and Pleats

Grafting Frusta

Modularized Folding: “Hill”

Examples:“Spike”

Examples:Negative Frusta

“Ripple”

“Volcano”

Composing Crease PatternsCPs of each “building block” CP for the combined whole

Pleats Split Creases

Partitioning into Sectors

Can’t unfold pleats one by one

“Cut” the paper into independent sectors then unfold pleats independently

Same as unfolding everything at once

Translating Sectors

Algorithm

Input: Tower of right frusta

Output: CP that folds into the tower

1. Start with a blank CP

2. Use bottommost frustum F for calculations

3. Divide the CP into sectors

4. Translate sectors outward according to F

5. Open up the pleats of F

6. Add the creases of F itself

7. Remove F from the tower

8. Repeat step 2—7 until the whole tower is consumed

Cheng, H. Y. & Cheong, K. H. Designing crease patterns for polyhedra by composing right frusta. Computer-Aided Design 44, 331-342 (2012)

http://www.sciencedirect.com/science/article/pii/S0010448511003058

GeneralizationNo need for n-fold symmetry!

General Frusta

Positive Frusta Negative Frusta

General Frusta

Positive Frusta Negative Frusta

Central Body and Pleats

Modified Sectors

Algorithm

Input: Tower of right frusta

Output: CP that folds into the tower

1. Start with a blank CP

2. Use bottommost frustum F for calculations

3. Divide the CP into sectors

4. Translate sectors outward according to F

5. Open up the pleats of F

6. Add the creases of F itself

7. Remove F from the tower

8. Repeat step 2—7 until the whole tower is consumed

ConclusionJust some final remarks…

Frustum Towers

Future: concave cross-sections? Many towers from one sheet?

Grafting

Modularized design algorithm Decrease-and-conquer origami design

Modularized folding algorithm Collapsing CPs is hard

Thank YouCHENG HerngYi herngyi.com

herngyi@mit.edu

mailto:herngyi@mit.edu