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A Man of Yet a Few More Words - by Swan Morrison

The Special Theory Of Egg Boiling

Introduction:

Despite its apparent simplicity, most cooks are daunted by the prospect of producing a perfectly boiled egg. This anxiety means that few, in the modern day, even make the attempt to do so.

It has been known since medieval times that two seemingly identical eggs, cooked in the same manner, could have very different consistencies when opened.

In earlier centuries, these variations had been attributed to the intervention of mischievous or malevolent spirits, and appropriate prayers were offered before and during the boiling process. Samuel Pepys noted in his diary of 1665, however, that such requests for divine assistance often appeared to have little effect on the culinary outcome!

It was not until the twentieth century that scientific progress finally led to recognition of the real causes of the problem: The inconsistency in egg cooking related to there being far more variables impacting on the process than had been previously recognised.

The physics, chemistry and mathematics which ultimately led to a solution to the problem of perfect egg boiling are unquestionably complicated, and this has, unfortunately, led to most egg boilings now being attempted only within academic and research institutions.

The purpose of this article is to summarise for a lay readership the theoretical issues involved, and provide a step by step guide to the practical process required to successfully boil an egg.

It is the earnest hope of all the academics who have contributed to the article that it will encourage many more cooks to develop the confidence to try it for themselves.

The variables that impact on the boiling of an egg:

1 - The egg, itself:

There are two inherent properties of an egg that impact on its behaviour when boiled: One is its size and the other is its freshness.

Quantifying these two properties has posed significant challenges to the finest scientific and mathematical minds.

The first major breakthrough occurred in 1968 when Professor Sandy Yoke of Oxford University was awarded the Nobel Prize for Mathematics for her resolution of what is known as the "Shape Profile Problem".

Isaac Newton in his Principia Mathematica had already deduced that all eggs are egg shaped. He had also concluded that for an egg of a specific volume, there are an infinite number of maximum diameters, maximum lengths and shell curvatures that can contain that same volume. These differences in geometry cause the passage of heat to differ through eggs of identical volume. This obviously affects their cooking properties.

Professor Yoke proved that the impact of egg geometry on cooking properties could be fully described by just two variables: the volume of the egg, usually denoted by the lower case letter, v, and its "shape profile coefficient", usually denoted by the lower case letter, s.

The second breakthrough came in 1976 when a postgraduate chemistry student at Cambridge University, John White, realised that the density of an egg related to its freshness.

Chemical changes gradually occur within the substance of an egg from the day it is laid. These ultimately result it being described as "bad". These changes also affect the boiling properties of the egg. John White not only calculated the relationship between freshness and density, but also between density and those boiling properties. He mathematically described this latter relationship with a coefficient he named the "culinary density" of the egg. This is usually denoted by the lower case letter, c.

The boiling properties that are directly related to inherent properties of an egg were, therefore, found to be fully described by its volume (v), its shape profile coefficient (s) and its culinary density (c).

Practical application of the above theory, however, had to await the development of an instrument that could accurately measure v, s and c. This came in the form of the magnetic image resonance or MRI scanner. After scanning an egg, these machines are able to directly provide values for these three variables.

Most household, of course, do not own an MRI scanner. Fortunately, one can be found in most hospitals, and the staff who operate such machines are often only too pleased to allow eggs to be scanned between the use of their machines for patients.

Tips:

When attending the hospital to have your eggs scanned, take a whole box of eggs. They can all be scanned while you are there, hence avoiding the need to return on every occasion that you are planning to boil an egg!

Don't forget to mark the v, s and c values on each egg in permanent marker as soon as it is scanned to avoid any later uncertainty as to which egg was which!

2 - Temperatures:

There are three temperatures, or sets of temperatures, that are relevant to the boiling of an egg. These are the temperature of the egg itself prior to boiling (t); the temperatures of the environments in which the egg is kept when not being boiled (e values), and the temperature of the water in which it is boiled. 

In the context of this article we are examining only the Special Theory of Egg Boiling. In the Special Theory, certain assumptions are made to simplify the theory and practice of producing a correctly boiled egg.

The key simplifications relate to environmental temperatures or e values.

The Special Theory assumes that:
- the room in which the egg is boiled is kept at a constant temperature prior to, and subsequent to, the boiling of the egg.
- the egg has remained in that environment for at least two hours prior to boiling.
- the egg is consumed either in that room or one kept at the same temperature, and that the egg does not experience any temperature changes in moving from the cooking to the eating room.
- the egg will be consumed as soon as its temperature has fallen to 45 degrees celsius.

The above simplifying assumptions are helpful because:
- the time taken for correct cooking depends on the temperature of the egg immediately prior to boiling. The mathematics is simplified if the egg is at the same temperature as that of its pre-boiling environment.
- an egg continues to cook, due to residual heat, when removed from the boiling water, and the extent of that additional cooking  relates to its rate of cooling. This, in turn, relates to the environmental temperature, together with the time and temperature at which the egg is finally consumed.

The General Theory of Egg Boiling takes account of changes in environmental temperatures at any stage of the process. It therefore includes multiple values of e (e₁, e₂, e₃ to e_n) together with durations within those environments (d_e1, d_e2, d_e3 to d_en). It also includes elements of differential calculus to take account of temperature gradients between successive environments.

The General Theory also contains terms to account for a temperature of an egg prior to boiling that differs from the pre-boiling environmental temperature.

The General Theory is, therefore, a powerful tool which allows a perfectly boiled egg to be consumed under any set of relevant variables. The massive increase in computational complication, however, makes it extremely cumbersome. Indeed, some would go as far as to say that it is impractical for everyday, household use.

Many people beileve that the boiling temperature of water is always 100 degrees celsius. This is not always the case, however, as the boiling temperature of water will be influenced by any impurities in the water together with factors that affect atmospheric pressure. The latter factors include height above sea level, the local acceleration due to gravity (variable across the Earth's surface because the planet is not a perfect sphere), and atmospheric conditions.

Fortunately the problem of impurities can be solved by boiling the egg in distilled water. All the other factors affect the one figure of barometric pressure (b) and this can be easily measured with a barometer at the location that the egg is boiled. There is no term for the boiling temperature of the water in the boiling time equation, below, as this is calculated from the barometric pressure.

Tips:

Clearly, using the Special Theory, e and t are equal and can both be measured by simply taking the temperature of the room in which the egg is to be cooked.

It is possible to buy a thermometer and barometer combined in one attractive, wall mounted display case. Why not get one for your kitchen?

3 - Personal preferences:

There is no one definition of a perfectly boiled egg because different people have different preferences in relation to hardness or softness.

The Special and General Theories of Egg Boiling are designed to produce consistency in relation to a defined preference. They do so by use of the hardness coefficient, h.

This is a number between 0 and 1. A zero value for h would result in an uncooked egg. A value of 1 would result in an egg cooked to the maximum level of hardness that it is possible for the constituents of an egg to reach.

Summary of variables used in the Special Theory of Egg Boiling:

The boiling time equation from the Special Theory of Egg Boiling :

T = h²vsc / b(20 + t + e)

Worked example using the typical values in the above table:

T = (0.6 X 3.14 X 3.14 X 5 X 520 X 10³ X 1.03) / (1033(20 + 20 + 20))

T = 15842405.28 / 61980 = 255.61 seconds or 4 minutes and 16 seconds

The step by step process for boiling an egg in accordance with the Special Theory of Egg Boiling:

The above theory and the resulting equation allow the exact boiling time required for any egg to be calculated. The cooking process must be standardised, however, to eliminate random variations.

This standardised process is as follows:

a - Have the egg scanned with an MRI scanner, as described above, to ascertain the values of v, s and c.
b - Write the v, s and c values on the egg in permanent marker as soon as it is scanned to avoid any later uncertainty as to which egg was which!
c - Ensure that the room in which the egg is to be cooked is maintained at a constant temperature.
d - Leave the egg in the room in which it is to be cooked for at least two hours prior to cooking in order that it attains the same temperature as that room.
e - Just prior to cooking, note the values of t and b.
f - Bring a pan of distilled water to the boil.
g - Use the boiling time equation to calculate the required egg boiling time, T.
h - Immerse the egg in the boiling water for exactly the period, T.
i - After removal from the boiling water, place the egg in an eggcup, ensuring that the environmental temperature surrounding the egg remains constant and at the pre-boiling value.
j - Monitor the temperature of the egg with a thermometer until it reaches 45 degrees celsius and then consume at once.

Conclusion:

The academics who have contributed to this article hope that the above explanations and guidance have demonstrated that boiling an egg is a task well within the capabilities of anyone with a basic knowledge of GCE mathematics.

We hope that it will encourage readers to obtain the limited amount of specialist equipment required and to try to boil eggs of their own. After all, what tastes better and what is healthier than a properly boiled egg?

Good luck and good dining!

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