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Soap Bubbles and Catenoids

Soap bubbles form such that they create a shape with the minimum surface area for the given constraints.  For a fixed volume the minimum surface area is a sphere, which is why soap bubbles will form spheres where possible.  We can also investigate what happens when a soap film is formed between 2 parallel circular lines like in the picture below: [Credit Wikimedia Commons, Blinking spirit]


In this case the shape formed is a catenoid – which provides the minimum surface area (for a fixed volume) for a 3D shape connecting the two circles.  The catenoid can be defined in terms of parametric equations:

Where cosh() is the hyperbolic cosine function which can be defined as:

For our parametric equation, t and u are parameters which we vary, and c is a constant that we can change to create different catenoids.  We can use Geogebra to plot different catenoids.  Below is the code which will plot parametric curves when c =2 and t varies between -20pi and 20 pi.

 

We then need to create a slider for u, and turn on the trace button – and for every given value of u (between 0 and 2 pi) it will plot a curve.  When we trace through all the values of u it will create a 3D shape – our catenoid.

Individual curve (catenary)


Catenoid when c = 0.1

Catenoid when c = 0.5

Catenoid when c = 1

Catenoid when c = 2

Wormholes

For those of you who know your science fiction, the catenoids above may look similar to a wormhole.  That’s because the catenoid is a solution to the hypothesized mathematics of wormholes.  These can be thought of as a “bridge” either through curved space-time to another part of the universe (potentially therefore allowing for faster than light travel) or a bridge connecting 2 distinct universes.

Screen Shot 2019-09-29 at 7.03.34 PM

Above is the Morris-Thorne bridge wormhole [Credit The Image of a Wormhole].

Further exploration:

This is a topic with lots of interesting areas to explore – the individual curves (catenary) look similar to, but are distinct from parabola.  These curves appear in bridge building and in many other objects with free hanging cables.  Proving that catenoids form shapes with minimum surface areas requires some quite complicated undergraduate maths (variational calculus), but it would be interesting to explore some other features of catenoids or indeed to explore why the sphere is a minimum surface area for a given volume.

If you want to explore further you can generate your own Catenoids with the Geogebra animation I’ve made here.

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IB Maths Exploration Guide

IB Maths Exploration Guide

A comprehensive 63 page pdf guide to help you get excellent marks on your maths investigation. Includes:

  1. Investigation essentials,
  2. Marking criteria guidance,
  3. 70 hand picked interesting topics
  4. Useful websites for use in the exploration,
  5. A student checklist for top marks
  6. Avoiding common student mistakes
  7. A selection of detailed exploration ideas
  8. Advice on using Geogebra, Desmos and Tracker.

Available to download here.

IB Exploration Modelling and Statistics Guide


IB Exploration Modelling and Statistics Guide

A 60 page pdf guide full of advice to help with modelling and statistics explorations – focusing in on non-calculator methods in order to show good understanding. Includes:

  1. Pearson’s Product: Height and arm span
  2. How to calculate standard deviation by hand
  3. Binomial investigation: ESP powers
  4. Paired t tests and 2 sample t tests: Reaction times
  5. Chi Squared: Efficiency of vaccines
  6. Spearman’s rank: Taste preference of cola
  7. Linear regression and log linearization.
  8. Quadratic regression and cubic regression.
  9. Exponential and trigonometric regression.

Available to download here.

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