block isoperimetric problems what do math and legos™ have in common? whales
TRANSCRIPT
![Page 1: BLOCK ISOPERIMETRIC PROBLEMS What do math and Legos™ have in common? Whales](https://reader035.vdocuments.us/reader035/viewer/2022062407/56649ced5503460f949ba970/html5/thumbnails/1.jpg)
BLOCK ISOPERIMETRIC PROBLEMS
What do math and Legos™ have in common? Whales.
![Page 2: BLOCK ISOPERIMETRIC PROBLEMS What do math and Legos™ have in common? Whales](https://reader035.vdocuments.us/reader035/viewer/2022062407/56649ced5503460f949ba970/html5/thumbnails/2.jpg)
Our Research Problem
Using Legos™, what is the smallest perimeter that will enclose a given area?
![Page 3: BLOCK ISOPERIMETRIC PROBLEMS What do math and Legos™ have in common? Whales](https://reader035.vdocuments.us/reader035/viewer/2022062407/56649ced5503460f949ba970/html5/thumbnails/3.jpg)
HISTORICAL MOTIVATION
Our Research Problem
Background Information
Whale Theorem Proof
Hydropronic Sequence
Future Work
![Page 4: BLOCK ISOPERIMETRIC PROBLEMS What do math and Legos™ have in common? Whales](https://reader035.vdocuments.us/reader035/viewer/2022062407/56649ced5503460f949ba970/html5/thumbnails/4.jpg)
Queen Dido
![Page 5: BLOCK ISOPERIMETRIC PROBLEMS What do math and Legos™ have in common? Whales](https://reader035.vdocuments.us/reader035/viewer/2022062407/56649ced5503460f949ba970/html5/thumbnails/5.jpg)
Previous Work
Maximum area is always square or pronic.
Perimeter is always even. Conclusion: A square or a square-like
rectangle has the greatest amount of area.
![Page 6: BLOCK ISOPERIMETRIC PROBLEMS What do math and Legos™ have in common? Whales](https://reader035.vdocuments.us/reader035/viewer/2022062407/56649ced5503460f949ba970/html5/thumbnails/6.jpg)
Relations Between The Projects They maximized area given a specified
perimeter. We wanted to minimize perimeter given a specified area.
Research Question: Given a specified area, what is the most efficient way to enclose that given area?
![Page 7: BLOCK ISOPERIMETRIC PROBLEMS What do math and Legos™ have in common? Whales](https://reader035.vdocuments.us/reader035/viewer/2022062407/56649ced5503460f949ba970/html5/thumbnails/7.jpg)
Definitions!
Lego™ Curve: a single continuous arrangement of 1x1 blocks such that each block has exactly two neighbors. *
Neighbor: a block that occupies the space adjacent to another block in one of the four cardinal directions along the grid. *
* Referenced from work of: Bower, Espinoza, Green, Roca, and Townsend
![Page 8: BLOCK ISOPERIMETRIC PROBLEMS What do math and Legos™ have in common? Whales](https://reader035.vdocuments.us/reader035/viewer/2022062407/56649ced5503460f949ba970/html5/thumbnails/8.jpg)
Improper LegoTM Curves
![Page 9: BLOCK ISOPERIMETRIC PROBLEMS What do math and Legos™ have in common? Whales](https://reader035.vdocuments.us/reader035/viewer/2022062407/56649ced5503460f949ba970/html5/thumbnails/9.jpg)
Definitions!
Perimeter: the number of blocks used to create a Lego™ curve. (Reminder: Perimeters are always even)
Area: the number of studs enclosed by a Lego™ curve.
![Page 10: BLOCK ISOPERIMETRIC PROBLEMS What do math and Legos™ have in common? Whales](https://reader035.vdocuments.us/reader035/viewer/2022062407/56649ced5503460f949ba970/html5/thumbnails/10.jpg)
WHALE THEOREM
Our Research Problem
Background Information
Whale Theorem Proof
Hydropronic Sequence
Future Work
![Page 11: BLOCK ISOPERIMETRIC PROBLEMS What do math and Legos™ have in common? Whales](https://reader035.vdocuments.us/reader035/viewer/2022062407/56649ced5503460f949ba970/html5/thumbnails/11.jpg)
The Whale Theorem
Using Legos™, what is the smallest perimeter needed to enclose a specified
area?
![Page 12: BLOCK ISOPERIMETRIC PROBLEMS What do math and Legos™ have in common? Whales](https://reader035.vdocuments.us/reader035/viewer/2022062407/56649ced5503460f949ba970/html5/thumbnails/12.jpg)
PROOF
Our Research Problem
Background Information
Whale Theorem Proof
Hydropronic Sequence
Future Work
![Page 13: BLOCK ISOPERIMETRIC PROBLEMS What do math and Legos™ have in common? Whales](https://reader035.vdocuments.us/reader035/viewer/2022062407/56649ced5503460f949ba970/html5/thumbnails/13.jpg)
Arrangement of BlocksArea: 45 Next Square or Pronic:
49
![Page 14: BLOCK ISOPERIMETRIC PROBLEMS What do math and Legos™ have in common? Whales](https://reader035.vdocuments.us/reader035/viewer/2022062407/56649ced5503460f949ba970/html5/thumbnails/14.jpg)
Arrangement of BlocksArea: 45 Next Square or Pronic:
49
![Page 15: BLOCK ISOPERIMETRIC PROBLEMS What do math and Legos™ have in common? Whales](https://reader035.vdocuments.us/reader035/viewer/2022062407/56649ced5503460f949ba970/html5/thumbnails/15.jpg)
Arrangement of BlocksArea: 45 Next Square or Pronic:
49
![Page 16: BLOCK ISOPERIMETRIC PROBLEMS What do math and Legos™ have in common? Whales](https://reader035.vdocuments.us/reader035/viewer/2022062407/56649ced5503460f949ba970/html5/thumbnails/16.jpg)
Arrangement of Blocks Area: 45 Next Square or Pronic:
49
![Page 17: BLOCK ISOPERIMETRIC PROBLEMS What do math and Legos™ have in common? Whales](https://reader035.vdocuments.us/reader035/viewer/2022062407/56649ced5503460f949ba970/html5/thumbnails/17.jpg)
Arrangement of BlocksArea: 45 Next Square or Pronic:
49
![Page 18: BLOCK ISOPERIMETRIC PROBLEMS What do math and Legos™ have in common? Whales](https://reader035.vdocuments.us/reader035/viewer/2022062407/56649ced5503460f949ba970/html5/thumbnails/18.jpg)
Arrangement of BlocksArea: 45 Next Square or Pronic:
49
![Page 19: BLOCK ISOPERIMETRIC PROBLEMS What do math and Legos™ have in common? Whales](https://reader035.vdocuments.us/reader035/viewer/2022062407/56649ced5503460f949ba970/html5/thumbnails/19.jpg)
Arrangement of Blocks Area: 45 Next Square or Pronic:
49
![Page 20: BLOCK ISOPERIMETRIC PROBLEMS What do math and Legos™ have in common? Whales](https://reader035.vdocuments.us/reader035/viewer/2022062407/56649ced5503460f949ba970/html5/thumbnails/20.jpg)
Arrangement of Blocks Area: 45 Next Square or Pronic:
49
![Page 21: BLOCK ISOPERIMETRIC PROBLEMS What do math and Legos™ have in common? Whales](https://reader035.vdocuments.us/reader035/viewer/2022062407/56649ced5503460f949ba970/html5/thumbnails/21.jpg)
Non Square & Non Pronic Formula In order to find the next possible square
or pronic use this formula:
Examples: A = 8 A = 2012
![Page 22: BLOCK ISOPERIMETRIC PROBLEMS What do math and Legos™ have in common? Whales](https://reader035.vdocuments.us/reader035/viewer/2022062407/56649ced5503460f949ba970/html5/thumbnails/22.jpg)
Examples
A = 8
A = 2012
![Page 23: BLOCK ISOPERIMETRIC PROBLEMS What do math and Legos™ have in common? Whales](https://reader035.vdocuments.us/reader035/viewer/2022062407/56649ced5503460f949ba970/html5/thumbnails/23.jpg)
Graph of: N
ext
Square
or
Pro
nic
Desired Area to Enclose
![Page 24: BLOCK ISOPERIMETRIC PROBLEMS What do math and Legos™ have in common? Whales](https://reader035.vdocuments.us/reader035/viewer/2022062407/56649ced5503460f949ba970/html5/thumbnails/24.jpg)
Graph of:
Perimeter
Are
a
![Page 25: BLOCK ISOPERIMETRIC PROBLEMS What do math and Legos™ have in common? Whales](https://reader035.vdocuments.us/reader035/viewer/2022062407/56649ced5503460f949ba970/html5/thumbnails/25.jpg)
HYDROPRONIC NUMBERS
Our Research Problem
Background Information
Whale Theorem Proof
Hydropronic Numbers
Future Work
![Page 26: BLOCK ISOPERIMETRIC PROBLEMS What do math and Legos™ have in common? Whales](https://reader035.vdocuments.us/reader035/viewer/2022062407/56649ced5503460f949ba970/html5/thumbnails/26.jpg)
Hydropronic Numbers
A number that is not square or pronic and can be arranged as a rectangle using its two closest factors to maintain minimum perimeter.
3, 8, 10, 15, 18, 21, 24, 28, 32, 35, 40, 45…etc
![Page 27: BLOCK ISOPERIMETRIC PROBLEMS What do math and Legos™ have in common? Whales](https://reader035.vdocuments.us/reader035/viewer/2022062407/56649ced5503460f949ba970/html5/thumbnails/27.jpg)
Hydropronic Numbers
Example: A = 24 Factors: 1, 2, 3, 4, 6, 8, 12, 24
Arrange a rectangle of 6 x 4
![Page 28: BLOCK ISOPERIMETRIC PROBLEMS What do math and Legos™ have in common? Whales](https://reader035.vdocuments.us/reader035/viewer/2022062407/56649ced5503460f949ba970/html5/thumbnails/28.jpg)
Hydropronic Area: 24Hydropronic: 6 x 4 (24)
![Page 29: BLOCK ISOPERIMETRIC PROBLEMS What do math and Legos™ have in common? Whales](https://reader035.vdocuments.us/reader035/viewer/2022062407/56649ced5503460f949ba970/html5/thumbnails/29.jpg)
Whale Theorem Area: 24Next Square or Pronic: 5 x
5 (25)
![Page 30: BLOCK ISOPERIMETRIC PROBLEMS What do math and Legos™ have in common? Whales](https://reader035.vdocuments.us/reader035/viewer/2022062407/56649ced5503460f949ba970/html5/thumbnails/30.jpg)
Optimal Perimeters for Area = 24
![Page 31: BLOCK ISOPERIMETRIC PROBLEMS What do math and Legos™ have in common? Whales](https://reader035.vdocuments.us/reader035/viewer/2022062407/56649ced5503460f949ba970/html5/thumbnails/31.jpg)
Finding Hydropronic Numbers
![Page 32: BLOCK ISOPERIMETRIC PROBLEMS What do math and Legos™ have in common? Whales](https://reader035.vdocuments.us/reader035/viewer/2022062407/56649ced5503460f949ba970/html5/thumbnails/32.jpg)
WhaLego Sequence
Numbers that can be arranged as rectangles and maintain the minimum perimeter are in the WhaLego sequence. All of these numbers can be classified as square, pronic, or hydropronic.
The first few numbers in this sequence are:
1, 2, 3, 4, 6, 8, 9, 10, 12, 15, 16, 18, 20, 21…etc
![Page 33: BLOCK ISOPERIMETRIC PROBLEMS What do math and Legos™ have in common? Whales](https://reader035.vdocuments.us/reader035/viewer/2022062407/56649ced5503460f949ba970/html5/thumbnails/33.jpg)
FUTURE WORK & REFERENCES
Our Research Problem
Background Information
Whale Theorem Proof
Hydropronic Numbers
Future Work
![Page 34: BLOCK ISOPERIMETRIC PROBLEMS What do math and Legos™ have in common? Whales](https://reader035.vdocuments.us/reader035/viewer/2022062407/56649ced5503460f949ba970/html5/thumbnails/34.jpg)
Future Work
Furthering Hydropronic Sequence Finding a generating formula
Lego Double Bubble Problem Enclosing two areas WhaLego sequence shows up
3-Dimensional Spaces
![Page 35: BLOCK ISOPERIMETRIC PROBLEMS What do math and Legos™ have in common? Whales](https://reader035.vdocuments.us/reader035/viewer/2022062407/56649ced5503460f949ba970/html5/thumbnails/35.jpg)
References
Brower, T.; Espinoza, J.; Green, A.; Roca, A.; Townsend, B. LEGOTM Isoperimetric Problem, preprint (2010).
Capogna, L.; Donatella, D.; Pauls, S.; Tyson, J. An Introduction to the Heisenberg Group and the Sub-Riemannian Isoperimetric Problem, 1st ed., Birkhauser Verlag AG, Basel, Boston, 2007.
![Page 36: BLOCK ISOPERIMETRIC PROBLEMS What do math and Legos™ have in common? Whales](https://reader035.vdocuments.us/reader035/viewer/2022062407/56649ced5503460f949ba970/html5/thumbnails/36.jpg)
Special thanks to the National Science Foundation (DMS- 1005046), Dr. Casey Douglas, and the St. Mary’s College of Maryland Department of Mathematics and Computer Science.
Questions?