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From its groundbreaking 1995 debut to its brand-new fourth instalment, the Pixar franchise has often been on the cutting edge of computer-generated animation. For that reason and many others, these films provide great stimuli for Science and Mathematics students of all ages, aiding discussion of concepts including randomisation, centripetal motion, flight and asset appreciation, as DAVE CREWE explains.

Released just under a quarter of a century ago, Toy Story (John Lasseter, 1995) was an experiment of sorts. This tale of what children's toys get up to when left to their own devices was Pixar's first foray into feature film and, most notably, was also the first feature film ever to be entirely computer-animated.

The experiment proved successful. Toy Story wowed critics

- even today, it has a 100 per cent positive critical response on Rotten Tomatoes (1) - and dominated the box office, becoming the highest-grossing film of 1995 in the United States. (2) For children of the 1990s, it was a defining film (I know I personally wore out my VHS copy through rewatches), and it went on to spawn three feature-length sequels - Toy Story 2 (Lasseter, 1999), Toy Story 3 (Lee Unkrich, 2010) and Toy Story 4 (Josh Cooley, 2019), the last of which recently cracked a casual billion US dollars at the global box office (3) - along with a handful of spin-offs, shorts and mountains of merchandise.

The Toy Story franchise, then, is perhaps the Platonic ideal of a Cinema Science subject. You can practically guarantee your students will be familiar with the films, while their family-friendly content ensures that they can be safely incorporated into your Science or Mathematics classroom at any year level. With modern-day streaming services, they are easily accessible - at the time of writing, the first three are available on Stan, though I suspect that by the time of publication they'll have parachuted over to the budding Disney+ service - whether you want to screen an entire film or specific clips.

There's plenty to explore across the four feature films in a scientific context, from how physics has enabled the evolution of computer animation, through the stunts seen in the films, to the financial mathematics associated with collectable toys. To infinity

- and beyond!

Advancing animation

Growing up in the 1980s and 1990s, I tended to associate cinematic sequels with a decline in quality. Budgets contracted, screenplay quality declined and stars were either replaced or looking worse for wear. Not so nowadays. Contrast, for instance, The Fast and the Furious (Rob Cohen, 200l) and The Fate of the Furious (F Gary Gray, 2017): (4) in the latter, the stunts are bigger, the budget's way bigger and the stars are fitter.

Toy Story is a great example of such franchise facelifts. Go back and watch the first film after finishing the latest instalment and you'll be shocked by the difference between the two. Animation that seemed revolutionary in 1995 now looks subpar compared to contemporary computer games, characterised by threadbare textures, simple lighting and some truly weird-looking humans. The character designs still hold up and the film is as engaging as it ever was, but it's hard to miss how much computer animation has improved in the intervening years.

To introduce this discussion in the classroom, I strongly recommend Insider's ten-minute video 'How Pixar's Animation Has Evolved over 24 Years, from Toy Story to Toy Story 4', (5) The breezily composed clip breaks down the differences in animation quality at either end of the franchise - for instance, contrasting the crudely rendered dog seen in Toy Story with the photorealistic cat featured in Toy Story 4 - with reference to the physical and computational challenges that needed to be overcome between the two films. As the video explains, a frame of Toy Story took 'from forty-five minutes to thirty hours to render'; nowadays, Pixar 'could render [the film] faster than you could watch the entire movie'. (6)

The biggest difficulty facing animators in the 1990s, however, was hair (or, in the case of the aforementioned pets, fur). For hair to appear believable, all of its thousands of strands need to be rendered and animated in a way that is physically realistic - a process that would have been too time-consuming to program by hand, but, equally, too computationally demanding for the technology of the day. Pixar cracked this problem in 200l's Monsters, Inc. (Pete Docter) with a splash of science:

Instead of looking at Sulley's [John Goodman] hair as a whole, they looked at each strand as a distinct particle. They had to look at every kind of force that would act on those particles, and thus how each one would move in reaction to them. So if you want to work at Pixar, you might need to know physics. (7)

Pixar went a few steps further with 2012's Brave (Mark Andrews & Brenda Chapman), modelling the mess of hair belonging to protagonist Merida (Kelly Macdonald) as '1,500 hand-placed, sculpted individual curls' in layers that 'varied the length, size and flexibility of each curl'. (8)

Practically, there's no way to have your students spend their Physics lessons sitting around sculpting hundreds of computer generated curls - nor, it must be said, is there much educational benefit. But this is an excellent framework that can act as a springboard for a cross-curricular activity aligning Science with some combination of Visual Art, Media Arts and/or Digital Technologies. Task your students with creating a realistic animation - whether via stop-motion, 3D computer animation or drawing by hand - demonstrating an understanding of the underlying mechanical calculations. Toys are a perfect subject here; even something as seemingly simple as modelling a toy like Woody falling to the floor is rich with conceptual complexity (which can be adjusted to suit different year levels).

There are other surprising applications of mathematics in computer animation. I was surprised to learn, for instance, that when Pixar's Finding Nemo (Andrew Stanton, 2003) needed to be converted to 3D in 2012, the studio 'found that certain aspects of the original could not be emulated in its new software':

The movement of seagrass, for instance, had been controlled by a random number generator, but there was no way to retrieve the original seed value for that generator. So animators manually replicated the plants' movements frame by frame, a laborious process. (9)

Not only is this, frankly, fascinating, but it also presents an excellent example of unexpected applications of mathematics, and an opportunity to unpack the mathematical underpinning of random number generators (and what it would mean to 'retrieve the original seed value').

* Why have hair and fur historically been so difficult to render with computer animation?

* Critically evaluate the claim that 'if you want to work at Pixar, you might need to know physics'.

* What is a 'seed value' in the context of a random number generator, and why is it important?

Carnival physics

The bulk of Toy Story 4 is set amid the colour and clamour of a small-town carnival. While the carnival setting is largely incidental to the story - it primarily provides a bright backdrop and the occasional obstacle for our miniature heroes - there's plenty on show for a Science lesson (or several).

A merry-go-round features prominently in the film. The reunited Woody (Tom Hanks) and Bo Peep (Annie Potts) survey the fairground from atop the ride, while later in the film they must undertake a perilous dash through the whirring machinery underneath. Merry-go-rounds are fertile ground for Physics teachers, allowing for the investigation of the centripetal forces associated with such rides. This could be done in a generic sense - using the film as stimulus material to zag off on a tangent, as befitting centripetal motion! - or your class could specifically analyse the merry-go-round seen in Toy Story 4. For example, the scene set beneath the ride threatens the toys with metal bars whirring past at astonishing speed; by timing their frequency, you could approximate the angular velocity of the merry-go-round and thus evaluate its safety (or lack thereof) as an amusement aimed at children by calculating the forces that a rider would experience.

You could also explore carnival games. Buzz Lightyear (Tim Allen) finds himself kept captive as a prize at one such game - alongside tethered toys Bunny (Jordan Peele) and

Ducky (Keegan-Michael Key) - over the course of the him. The design of carnival games is a careful art; they shouldn't be so hard that players have no chance of winning the cheaper prizes, but, equally, the especially valuable prizes should be nigh on impossible to win. Task an upper primary or junior secondary Science class with building their own carnival game, and they'll have an opportunity to practise design, mathematics and physics skills without even realising it.

* By examining the associated centripetal forces, suggest the best speed (in terms of angular velocity) for a merry-go-round aimed at young children.

* What physics concepts are associated with the other carnival rides seen in Toy Story 4?

Falling... with style

The hook for the first Toy Story film revolves around Buzz Lightyear's identity crisis. The brand-new action figure is convinced he's a 'space ranger' - a member of a sort of intergalactic police force - rather than a toy, and the central conflict derives from the disagreements that result between him and Woody. Their main point of contention? Buzz's ability to fly. Equipped with retractable plastic wings, Buzz isn't a flying toy, (10) but he doesn't let that stop him; through some convenient leaps and bounces, he's able to convince the toys of his flying ability early in the film. Well, all of the toys bar Woody, who sniffs at the stunt as merely 'falling with style'.

Perfect opportunity for more physics here. Plenty of items that seemingly 'fly' - from paper planes to hang-gliders - are really just 'falling with style', so there's an opportunity in this to explore the science behind such aerodynamics. A simple experiment could be to research and design the best possible paper planes, investigating how shape and materials have an effect on flight times and how these principles could be related to 'flying' toys or gliding apparatuses.

There's also an opportunity to tie the discussion around Buzz's flying ability (or lack thereof) to theme-park physics, branching off from the preceding 'carnival physics' section. Buzz's 'falling with style' demonstration involves him launching from a rollercoaster-style toy race-car set, speeding around a loop-the-loop in the racetrack before being propelled into the air. While this scene might be a flimsy premise in and of itself, if paired with an investigation of the physics associated with carnival rides it presents a great opportunity to explore the physics of mechanical energy and circular forces associated with roller-coasters. In particular, you could investigate if it's realistic for Buzz (or Joan Cusack's Jessie, who performs an almost identical stunt at the end of Toy Story 2) to get as much air as he does after leaving the racetrack. This doesn't have to be entirely theoretical, either; pick up a toy roller-coasterstyle race-car set and launch an action figure from it!

Indeed, one of the advantages of using the Toy Story franchise as scientific stimulus material is the easy ability to generate experiments using to-scale toys rather than miniaturised simulacra, as you would need to if seeking to replicate a stunt from a live-action film. Case in point: order a Buzz Lightyear toy that actually flies - thanks to drone-esque propellers - from the Disney shop," and use it to explore the physics associated with such flying toys (any drone will do in a pinch). Alternatively, build on the aforementioned paper-plane experiment by providing your students with (reasonably priced) action figures and asking them to modify them into flying toys - or, at least, toys that can fall in style.

* Is it realistic that a toy shaped like Buzz Lightyear would be able to fly as seen in the first act of Toy Story (the scene that runs from 0:17:40 to 0:19:08)? Why or why not?

* What scientific principles are important when designing 'flying' devices like paper planes or hang-gliders?

Collector's items

Toy Story 2 reveals that Woody isn't some random cowboy toy, but a rare collector's item from an old puppet TV show called Woody's Roundup. After he is abducted by collector Al (Wayne Knight), the film makes a point of underlining Woody's value as a collectable, especially when part of a complete set with his Woody's Roundup 'co-stars'.

This isn't completely concocted by the screenwriters. Plenty of real-world toys attract sale prices with plenty of zeroes. The classic example is that of the original Star Wars figurines, which can attract prices into the tens of thousands of dollars (provided they're in mint condition, naturally). (12) Toy museums - like the one that Al plans to sell Woody to - do exist. If you're teaching History, you could use this to emphasise the breadth and diversity of how we might define a historical relic ... but this is Cinema Science, so let's instead get our hands dirty with the complicated commerce underpinning such collectables.

Depending on the requirements of the curriculum, a Mathematics teacher can take this stimulus in a number of different directions. Once you've laid out your assumptions about how collectables accrue value, the worth of a given collectable can be modelled using arithmetic or geometric sequences. Neither model is likely to be perfectly realistic - even the most reliable collectable item doesn't tend to go up in value by a set amount or percentage each year - which provides an excellent opportunity to discuss the assumptions associated with these mathematical models and identify the limitations of applying such models to life-related contexts.

You could also explore the changing values of collectable toys in the more specific context of financial mathematics. Any unit exploring taxation will require students to understand the concept of appreciation and depreciation of assets; why not do so by exploring how collectable toys' value changes over time? This could be an illustrative example or, if you have the time, an extended modelling task in which students record the historical value of such toys and attempt to approximate their rate of appreciation or depreciation into the future.

* What's the most valuable collectable toy, and at what rate has its value increased?

* Do you need to declare the increasing value of a collectable - like a rare toy - on your tax return in Australia?

Odds and ends

Rocket science! Forever valorised as the most difficult branch of science, it is an ever-enticing one, too: there's something ineffably appealing about tinkering with rockets. Toy Story doesn't really get into the science of rockets - even if one suspects that prepubescent sociopath Sid (Erik von Detten), who delights in destroying toys with his own arsenal of bombs and miniature missiles, has some understanding of the technicalities - but that shouldn't stop us. While I wouldn't recommend launching toys from rockets in your Science classroom (per Woody's advice: 'Rockets explode!'), why not task your students with building the most successful rocket they can make from household materials, like the classic lemon-juice rocket? (13)

Toy Story 3's most memorable moment is when our beloved toys face seemingly certain doom in a garbage incinerator. Though they're thankfully ultimately rescued from their fate, their forays into the depths of a garbage dump - through landfill, past shredders, across conveyor belts - allow for an investigation into the garbage-disposal ecosystem. How do we manage our trash in a sustainable and environmentally considerate way? (And, yes, in many cases the answer to that question is: we don't.)

Across the four films, there's plenty of further stimuli for the eager Science student. Maybe you could explore Keanu Reeves' Duke Caboom, a Canadian daredevil whose jumping stunts leave a little to be desired, in the context of projectile motion. A unit on electricity could incorporate the operation of toys' voice boxes, which proves a central narrative element of Toy Story 4. But, instead, I'll leave you with an example of Toy Story's influence on science, rather than the other way around.

In 2008, Buzz Lightyear travelled into infinity - and space. Accompanying commander Mark Kelly, pilot Kenneth Ham and a crew of five mission specialists on the NASA space shuttle STS-124 was an action figure of Buzz. The toy was used to engage young students and even took part in some zero G experiments (while, naturally, providing some handy marketing for Disney).'" Sadly, the educational games designed at the time are no longer available online, (15) but the story is nonetheless a great way to engage younger students in the possibilities of astrophysics.

Dave Crewe is a Brisbane-based secondary teacher and critic.


(1) 'Toy Story', Rotten Tomatoes, <>, accessed 24 August 2019.

(2) '1995 Domestic Grosses', Box Office Mojo, <>, accessed 24 August 2019.

(3) 'Toy Story 4', Box Office Mojo, <>, accessed 24 August 2019.

(4) These films were the focus of the second ever Cinema Science piece; see David Crewe, 'Cinema Science: The Fast and the Furious and the Mechanics of Dangerous Driving', Screen Education, issue 87, 2017, pp. 32-9.

(5) Kyle Desiderio & Ian Phillips, 'How Pixar's Animation Has Evolved over 24 Years, from Toy Story to Toy Story 4', Insider, 20 June 2019, <>, accessed 24 August 2019.

(6) ibid.

(7) ibid.

(8) Claudia Chung, quoted in Emilie Lorditch, 'Brave Features Hair-raising Animations', Inside Science, 20 June 2012, <>, accessed 24 August 2019.

(9) Marty Perlmutter, 'The Lost Picture Show: Hollywood Archivists

Can't Outpace Obsolescence', IEEE Spectrum, 28 April 2017, <>, accessed 24 August 2019.

(10) Indeed, he suffers a crisis of confidence around the midpoint of Toy Story when he sees a Buzz Lightyear advertisement prominently featuring the caveat that Buzz Lightyear is 'Not a Flying Toy'.

(11) Available at <>, accessed 14 September 2019.

(l2) See Maude Campbell, 'The Most Expensive Star Wars Toys', Popular Mechanics, 7 March 2019, <>, accessed 3 September 2019.

(13) For a guide on how to make these, see 'Acids and Bases: Testing Rockets', PBS LearningMedia, <>, accessed 3 September 2019.

(14) See 'NASA Launches Disney's Buzz Lightyear on Space Shuttle',, 1 June 2008, archived at <https://web.archive.0rg/web/20080701203519/>; and 'Mission Information: STS-124', NASA website, <>, both accessed 3 September 2019.

(14) You can find the links at the following address, but unfortunately they're all broken at the time of writing and unlikely to be updated: 'Buzz Lightyear's Dream Come True', NASA website, 31 May 2008, <>, accessed 25 August 2019.
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Date:Dec 1, 2019

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