**Objectives
**Students will:

Learn algebraic-geometric patterns for how changes in size affects surface area, volume and mass.

Be able to do numerical calculations for such changes

See how real world materials (e.g. bone) stand up to changes in scale.

### Giant monster movies – **Kaijū** (怪獣 ) Japanese for “strange beast” – are a fun genre of cinema.

**Kaijū**

### Godzilla, King Kong, Pacific Rim and Cloverfield. We see beasts of incredible size and mass

### Live-action giant robot film is known as Tokusatsu (特撮)

### How does changing the size of a creature affect it’s surface area, it’s volume, and it’s mass?

### What real-world limiting factors will appear? These are questions of *scaling*.

__________________________________________

The following is adapted from, Fran Bagenal, Astrophysical and Planetary Sciences, University of Colorado – http://lasp.colorado.edu/~bagenal/MATH/AreaVolume.html

**Proportionality and the Value of Ratios**

### Consider a square – length “D” on a side:

### The area is D x D = D^{2} (with the appropriate units – meters^{2} or km^{2} )

### Now double the length of each side (each side is now 2D)

### Area = 2D x 2D = (2D) ^{2} = 4 x D^{2} (with same units).

### The** ratio of sizes** of the squares

### = (Size Big)/(Size Small) = (2D)/(D) = 2.

### The **ratio of the areas** of the squares

### = (Area Big)/(Area Small) = (2D)^{2} / D^{2} = 2^{2} = 4.

### Next , let’s look at **volume.** Consider a volume of a cube that is D on a side.

### Volume of cube = D x D x D = D^{3 } (units are meters^{3} or km^{3} or feet^{3} )

### Next consider a cube that is twice the size – 2D on a side. T

### Volume = (2D)^{3} = 2^{3} D^{3} = 8 D^{3}.

### Taking** ratios **we can summarize :

### (Size Big)/(Size Small) = (2D)/ D = 2 -> We double the size

### (Area Big)/(Area Small) = (2D)^{2}/ D^{2} = 2^{2} = 2 x 2 = 4 = [(Size Big)/(Size Small)]^{2}

### -> We get four times the surface area.

### (Volume Big)/ (Volume Small) = (2D)^{3}/ D^{3} = 2^{3} = 2 x 2 x 2 = 8 = [(Size Big)/(Size Small)]^{3}

### -> We get 8 times the volume.

### – – – – – –

### What about cells, or planets and stars? They are spheres. Let’s start with circles: Begin with a circle of radius R.

### Area =p·R^{2}.

### Now double the size of the circle – double the radius (and/or double the diameter)

### Area = p (2R)^{2} = = p ·2^{2} R^{2} = π· 4 R^{2}

### Now move from 2 dimensions to 3 dimensions: from circles to spheres.

### Area of a sphere = 4p R^{2}

### Now double the size (double length of the radius)

### Area = 4p (2R)^{2} = 4p 2^{2}R^{2} = 16 pR^{2}

### Volume (of smaller sphere) = 4/3 pR^{3}

### Volume (larger sphere) = 4/3 p(2R)^{3 }= 4/3 p2^{3}R^{3 }= 4/3 x 8 pR^{3}= 32/3 pR^{3}

### Taking** ratios** we get

### (Area Big Circle)/ (Area Small Circle) = (p (2R)^{2}) / ( p R^{2}) = 2^{2}= 4

### (Area Big Sphere)/ (Area Small Sphere) = (4p (2R)^{2}) / ( 4p R^{2}) = 2^{2}= 4

### (Volume Big Sphere)/ (Volume Small Sphere) = (4/3 p (2R)^{3}) / ( 4/3 p R^{3}) = 2^{3}= 8

_______________________________

### What about irregular shapes? Will the math be more complex – or easy What’s the formula for the area of a frog?

### Area of a frog = “something” x H^{2}

### Now, what happens if the frog is twice as big?

### Area of big frog = “something” x (2H)^{2}= “something” x 2^{2}H^{2} = “something” x 4 H^{2}

### Volume of a frog = “something else” x H^{3}

### Volume of the big frog

### = “something else” x (2H)^{3 }

### = “something else” x 2^{3}H^{3 }

### = “something else” x 8 H^{3}

### We have no idea what the “something else” is.

But guess what? Doesn’t matter. Take ratios!

### (Area Big Frog)/ (Area Small Frog) = (something x (2H)^{2}) / (something x H^{2}) = 2^{2}= 4

### (Volume Big Frog)/ (Volume Small Frog) = (something else (2H)^{3}) / (something else H^{3}) = 2^{3}= 8

### Conclusion? All you need to think about is the ratio.

### Ratio of areas is always (Area Big)/ (Area Small) = [(Size Big)]/(Size Small)]^{2}.

### Ratio of volumes is always (Volume Big)/(Volume Small) = [(Size Big)]/(Size Small)]^{3}.

**Area** is** proportional** to **Size**^{2}

^{2}

**Volume** is **proportional** to **Size**^{3}

^{3}

### This works for squares, spheres, frogs,gorillas (King Kong!), dinosaurs (Godzilla), etc.

_________________

# Godzilla vs Scaling Laws of Physics

Adapted from “Godzilla Versus Scaling Laws of Physics,”

Thomas R. Tretter, University of Louisville, Louisville, KY

The Physics Teacher Vol.43, Nov. 2005

### By the 17th century the importance of scaling effects on structural strength was noted by Galileo (Dialogues Concerning Two New Sciences, 1638)

### A horse falling eight feet would break many bones

### A dog falling eight feet would walk away, usually with no broken bones.

### A dog could carry two or three similarly sized dogs on its back.

### A horse could not carry even one similarly sized horse.

### __________________________________________

# Godzilla, 1998, with Matthew Broderick.

View scene at 10:15 – footprint

### The scale of the image is estimated from the size of a case that Broderick is carrying: the case is approximately the same length as his back, from waist to shoulder.

### On screen (measured with ruler):

### case length = 2 cm

### foot length = 28 cm, foot width = 23 cm

### Real length of a person’s back = about 52 cm

### case length = about 52 cm

### Therefore 1 cm on screen = 26 cm in real life.

### Length of foot = 26 * 28 cm = 728 cm

### Width of foot = 26 * 23 cm = 598 cm

### Approximating her feet as rectangular,

### area of Godzilla’s 2 feet = (728 cm)(598 cm)(2) = 870,688 cm2

– – – – –

### Bone strength is proportional to cross section area of the bone

### Major source of stress on the leg bones: the weight being supported.

### Let’s find the stress pressure exerted on our own leg bones.

### Measure the circumference of an ankle, then find the cross sectional area of your two ankles together.

### Ankle circumference C = 25 cm = 2πr

### r = 25 cm / (2π) = 4.0 cm

### Area = πr2 = 50 cm2

### For two ankles, area = 100 cm2

### Average person’s weight (presses down on ankles) = 165 lb

### Convert to metric units:

### 1 pound = 4.45 newtons -> 165 lb = 734 N

### Pressure = Force / area = 734 N /100 cm2 = 7.3 N/cm2

### Now let’s do this for an elephant: Foot circumference = 52 in (130 cm)

### Weight = 9300 lb = 4.1 x 10^4 N

### Results in 7.5 N/cm2 , pressure exerted by each foot.

### Pressures are similar across species regardless of size.

### Conclusion? Pressures are related to the compressive strength of bone material.

### They can’t be greatly exceeded regardless of how large the animal may be.

### 26 minutes: Godzilla stomps NYC

### Estimate the area of Godzilla’s ankles, as compared to the area of her feet

### Ankle is, at best, about 1/4 the area of the foot

### Using previous data, estimate 2.2 x 10^5 cm2 for total cross sectional area of the two ankles.

### Estimate Godzilla’s weight.

____________________

34:00 Post rampage scenes including a building with a huge hole through it

### Size of this hole is used to estimate that Godzilla’s volume is 8.5 x 10^4 m3 (see Fig. 2).

Assume that Godzilla’s density is approximately that of water, 1000 kg/m3, results in a weight of 8.3 x 10^8 N.

### This value together with the ankles’ cross sectional area gives a pressure of about 3.8 x 10^3 N/cm2!

### This is about 500 times greater pressure on the bone than for other creatures.

## Learning standards

**2016 Massachusetts Science and Technology/Engineering Curriculum Framework**

Appendix VIII Value of Crosscutting Concepts and Nature of Science in Curricula

In grades 9–12, students can observe patterns in systems at different scales and cite patterns as empirical evidence for causality in supporting their explanations of phenomena. They recognize that classifications or explanations used at one scale may not be useful or need revision using a different scale, thus requiring improved investigations and experiments. They use mathematical representations to identify certain patterns and analyze patterns of performance in order to re-engineer and improve a designed system.

**Next Gen Science Standards HS-PS2 Motion and Stability**

Crosscutting Concepts: Different patterns may be observed at each of the scales at which a system is studied and can provide evidence for causality in explanations of phenomena. (HS-PS2-4)

**A Framework for K-12 Science Education**

Scale, proportion, and quantity. In considering phenomena, it is critical to recognize what is relevant at different measures of size, time, and energy and to recognize how changes in scale, proportion, or quantity affect a system’s structure or performance….

The understanding of relative magnitude is only a starting point. As noted in Benchmarks for Science Literacy, “The large idea is that the way in which things work may change with scale. Different aspects of nature change at different rates with changes in scale, and so the relationships among them change, too” [4]. Appropriate understanding of scale relationships is critical as well to engineering—no structure could be conceived, much less constructed, without the engineer’s precise sense of scale.

- Dimension 2, Crosscutting Concepts, A Framework for K-12 Science Education: Practices, Crosscutting Concepts, and Core Ideas (2012)

## Film clips

Antman Thomas The Tank Engine fight scene

Cloverfield – bomb the creature

Fantastic Voyage | #TBT Trailer | 20th Century FOX

Pacific Rim – scene girl in street

Ghostbusters – The Stay Puft Marshmallow Man

Ghostbusters 2 Stature of Liberty

Godzilla 2014 – out of the water

Godzilla – all sightings analysis

(1964) Godzilla, Rodan, Mothra vs Ghidrah

King Kong battles T Rex – original version

2:12 / 3:53 Sky Captain and the World of Tomorrow clip

Transformers 1 Movie – Base Attack

## Megastructures

Extreme Engineering S1E09B Boston Big Dig

Boston Big Dig | Megastructures – National Geographic

Mega Structures – Burj Khalifa, Dubai [National Geographic]

Megastructures – Aldar HQ Abu Dhabi Worlds First Round Skyscraper Documentary National Geographic.

Deep Sea Drillers | Megastructures National Geographic Factories

[…] Finding areas and volumes […]

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