# Fermat Spirals for Layered 3D Printing | Two Minute Papers #77

## Метаданные

- **Канал:** Two Minute Papers
- **YouTube:** https://www.youtube.com/watch?v=6rNcAVr-U4s
- **Дата:** 05.07.2016
- **Длительность:** 4:17
- **Просмотры:** 7,595
- **Источник:** https://ekstraktznaniy.ru/video/14806

## Описание

The paper "Connected Fermat Spirals for Layered Fabrication" is available here:
http://irc.cs.sdu.edu.cn/html/2016/2016_0519/222.html

The ThatsMaths article on sunflowers + paper "Fibonacci patterns: common or rare?" is available here:
https://thatsmaths.com/2014/06/05/sunflowers-and-fibonacci-models-of-efficiency/
http://www.sciencedirect.com/science/article/pii/S2210983813001314

Another nice application of Hilbert curves for spatial indexing (thanks for the link TheJonManley!):
http://blog.notdot.net/2009/11/Damn-Cool-Algorithms-Spatial-indexing-with-Quadtrees-and-Hilbert-Curves

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The thumbnail backgro

## Транскрипт

### Intro []

dear fellow Scholars this is 2minute papers with car what are Hilbert curves are repeating lines that are used to fill a square such curves so far have enjoyed applications like drawing zigzag patterns to prevent biting in our tail in a snake game or Jokes Aside it is also useful in for instance choosing the right pixels to start tracing rays of light in light simulations or to create good strategies in assigning numbers to different computers in a network these numbers by the way we call IP addresses these are

### Examples [0:37]

just a few examples and they show quite well how a seemingly innocuous mathematical structure can see applications in the most mindbending ways imaginable so here's one more

### Fermat Spiral [0:48]

actually two more Fermat's spiral is essentially a long line as a collection of low curvature spirals these are generated by a remark ably simple mathematical expression and we can also observe such shapes in modern nature for instance in a sunflower and the most natural question emerges in the head of every seasoned fellow scholar why is that why would nature be following mathematics or anything to do with what fermat wrote on a piece of paper once it has only been relatively recently shown that as the seeds are growing in the sunflower they exert forces on each other that therefore they cannot be arranged in an arbitrary way we can write up the mathematical equations to look for a way to maximize the concentration of growth hormones within the plant to make it as resilient as possible in the meantime this Force exertion constraint has to be taken into consideration if we solve this equation with Blood Sweat and Tears we may experience some moments of great Peril but it will be all washed away by the beautiful sight of this arrangement M this is exactly what we see in nature and which happens to be almost exactly the same as a mind-bendingly simple firit spiral pattern words fail me to describe how amazing it is that mother nature is essentially able to find these solutions by herself really cool isn't

### Conclusion [2:19]

it if our mind wasn't blown enough yet firit spirals can also be used to approximate a number of different shapes with the added constraint that we start from a given point take an enormously long journey of low curvature shapes and get back to almost exactly where we started this again sounds like an innocuous little game evoking ill-concealed laughter in the audience as it is presented by as excited as underpaid mathematicians however as always this is not the case at all researchers have found that if we get a 3D printing machine and create a layered material aterial exactly like this the surface will have a higher degree of fairness be quicker to print and will be generally of higher quality than other possible shapes if we think about it if we wish to print a prescribed object like this cat there is a stupendously large number of ways to fill this space with curves that eventually form a cat and if we do it with firment spirals it will yield the highest quality print one can do at this point in time in the paper this is demonstrated for a number of shapes of varying complexities and this is what research is all about finding interesting connections between different fields that are not only beautiful but also enrich our everyday lives with useful inventions in the meantime we have reached our first Milestone on patreon and I am really grateful to you fellow Scholars who are really passionate about supporting the show we are growing at an extremely rapid pace and I am really excited to make even more episodes about these amazing research works thanks for watching and for your generous support and I see you next