The Mathematics. Growth, Form and Size. Part 1

Transcription

1 The Mathematics of Growth, Form and Size Part 1 Would you rather be big or small?

2 Why is an elephant shaped like an elephant?

3 Some answers: (1) its parents were like that (DNA) genetic (2) its lifestyle and environment demand it ecological (3) complex biological processes over long periods biological/evolutionary (4) the mathematical principles of the universe mathematical (5) the Creator created elephants that way makes a Space-Time universe to make elephants that way theological

4 It couldn't look like a scaled-up antelope, and you couldn't have a scaled-down elephant! Size and shape are intimately connected by mathematical principles

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8 Scaling lengths, areas, volumes length area length x 10 bonestrength x 100 volume 1 Weight x

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11 2cm Linear scale: Trial footlength L: Foot area A: Total weight W: Foot pressure (W/A):

12 Linear scale: Trial footlength L: Foot area A: Total weight W: Foot pressure (W/A):

13 Linear scale: Trial footlength L: Foot area A: Total weight W: Foot pressure (W/A):

14 Linear scale: Trial footlength L: Foot area A: Total weight W: Foot pressure (W/A):

15 Linear scale: Trial footlength L: Foot area A: Total weight W: Foot pressure (W/A): Proper footlength? To get W/A = 0.25 =1/4 A = 4W L = 2 W

16 Linear scale: Trial footlength L: Foot area A: Total weight W: Foot pressure (W/A): Proper footlength (approx.): To get W/A = 0.25 =1/4 A = 4W L = 2 W

17 2cm

18 Dinosaurs drawn to different scales: put them in order of size.

19 Dinosaurs drawn to different scales: put them in order of size. Diplodocus 28m long Brachiosaurus 26m by 12 m, tons Comsognathus 60cm by 30cm, 3 kg chicken-ike Spinosaurus 15m long Ultrasaurus 30m long, 12 m high 80 tons Ceratosaurus 5m long

20 What can we deduce about shape and size and behaviour of living things? Elephants? Tyrannosaurus Rex? Springbok, cats, mice, grasshoppers? Lion paws, puppy paws, women's feet? How much can you carry? Why are my feet sore?

21 Diversion: What if an elephant fell out of an aircraft? And a mouse? Question: What would you like to be in your reincarnation: BIG? SMALL?

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25 Which is better to be: BIG? SMALL? BIG SMALL GOOD for being stronger, reaching higher, travelling further, faster Bad vulnerable to being trodden on, to fire, flood, etc..

26 Which is better to be: BIG? SMALL? BIG SMALL GOOD for being stronger, reaching higher, travelling further, faster Bad vulnerable to being trodden on, to fire, flood, etc.. Bad for running: bones will break, and in falling. GOOD for running, jumping, falling

27 Which is better to be: BIG? SMALL? BIG SMALL GOOD for being stronger, reaching higher, travelling further, faster Bad vulnerable to being trodden on, to fire, flood, etc.. Bad for running: bones will break, and in falling. GOOD for running, jumping, falling Bad need long tongue/wrinkled skin/big ears, convoluted intestinal tracts and lungs, expanded trachea. Big things struggle in deserts with some exceptions! GOOD for keeping cool, absorbing oxygen, digesting food. Surface services volume: S/V prop. to 1/h.

28 Which is better to be: BIG? SMALL? BIG SMALL GOOD for being stronger, reaching higher, travelling further, faster Bad vulnerable to being trodden on, to fire, flood, etc.. Bad for running: bones will break, and in falling. GOOD for running, jumping, falling Bad need long tongue/wrinkled skin/big ears, convoluted intestinal tracts and lungs, expanded trachea. Big things struggle in deserts with some exceptions! GOOD for keeping cool, absorbing oxygen, digesting food. Surface services volume: S/V prop. to 1/h. GOOD for keeping warm: heat transfer prop. to 1/d. Bad have to eat constantly, little things struggle in polar regions...

29 BIG SMALL Bad for survival in deserts with some exceptions! GOOD for survival in deserts

30 BIG SMALL Bad for survival in deserts with some exceptions! GOOD for survival in deserts Bad ponderous, slow and clumsy: accel. = F/m prop. to 1/d GOOD for making quick movements

31 BIG SMALL Bad for survival in deserts with some exceptions! GOOD for survival in deserts Bad ponderous, slow and clumsy: accel. = F/m prop. to 1/d GOOD for making quick movements GOOD for getting wet. Bad for getting wet: weight increase prop.to 1/d

32 BIG SMALL Bad for survival in deserts with some exceptions! GOOD for survival in deserts Bad ponderous, slow and clumsy: accel. = F/m prop. to 1/d GOOD for making quick movements GOOD for getting wet. Bad for getting wet: weight increase prop.to 1/d Very Bad for getting body down and up again: Work = Fd prop. to d4 GOOD for drinking, lying down...

33 So that's why an elephant is shaped like it is!

34 Conclusion Small is better? But Nature is infinitely resourceful and generally finds a way, or many ways Things don't outgrow their natural limits unless humans get involved and then we have the responsibility of choice! The size and shape and behaviour of an animal, organism or community, are intimately connected, in a delicately adjusted, constantly adapting balance of many mechanisms.

35 Exercises 1. Why has a giraffe got a long neck? 2. Can a small dog jump as high as a large dog? (Or a small cat/antelope jump as high as a large one?) 3. What size is best for survival in hot deserts? What size is best if we incorporate water-storage ability? 4. What would be the consequences of doubling the size of a passenger airliner? (Think about income, wing-size, fuel, compare albatross and humming bird) 5. What would be the consequences of doubling the size of luxury cruise ship? What brilliant insight led to the construction of the first steamship to cross the Atlantic?

36 1. Why has a giraffe got a long neck?

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38 2. Can a small cat jump as high as a large cat?

39 3. What size is best for survival in hot, dry deserts? What if we incorporate water-storage?

40 4. What would be the consequences of doubling the size of a passenger airliner?

41 5. What would be the consequences of doubling the size of luxury cruise ship? What brilliant insight led to the construction of the first steamship to cross the Atlantic?

42 Apatasaurus Diplodocus 28m long

43 Supersaurus 40m long, 100 tons

44 Ultrasaurus 30m long, 12 m high 80 tons

45 Apatasaurus

46 Brachiosaurus 26m by 12 m, tons

47 Ceratosaurus 5m long Spinosaurus 15m long

48 Abelisaurus Baryonyx 9 m long Suchomimus 9 m long 11 m long

49 Comsognathus 60cm by 30cm, 3 kg chicken-ike

50 Tyrannosaurus rex Triceratops Stegosaurus Triceratops Allosaurus Anklyosaurus Oviraptor 6m

51 theory theory