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One of my hobbies is sacred geometry‭ ‬-‭ ‬loosely defined,‭ ‬it’s the study and use of mathematical archetypes in nature and culture,‭ ‬often with a focus on traditional compass and straightedge constructions.‭

Don’t worry‭; ‬I’m not about to go off into numerology,‭ ‬telling you that you can derive the groovy cosmic secrets of the ancients by studying numeric coincidences.‭ ‬I actually take a rather dim reading spurious meaning into special cases of the Interesting Number Theorem.‭* ‬The worst case scenario might resemble the Arronofsky film‭ “‬Pi‭”‬,‭ ‬except without the sweet soundtrack.

Nonetheless,‭ ‬exploring the traditions behind sacred geometry can give insight into art and design.‭ ‬The traditional compass and straightedge drove the development of mathematics until just a century or two ago,‭ ‬and modern algebra has its roots in questions about why it is,‭ ‬exactly,‭ that ‬you can’t construct a perfect heptagon‭. The practice is also fun and relaxing, and can even give occasional insights into physical reality.

One subject that comes up a lot in these discussions is a number called the golden ratio,‭ ‬sometimes abbreviated phi.‭ ‬Phi is defined in terms of the relationship between a line segment and its parts.‭

What’s Up with the Bad Chemistry?

Watts Up With That is a very, very silly website.

Here’s what I mean: In a recent article at WUWT, chemical engineering graduate Steve Burnette tries to dismiss concerns about ocean acidification, but his claims are outright wrong when they are coherent. The centerpiece of the article is a calculation meant to estimate the change in ocean pH over the 20th century. Unfortunately, it’s been badly bungled.

First, let’s go over a few preliminaries.

Figure 1: The carbonic acid equilibrium system. Atmospheric CO2 is in equilibrium with dissolved CO2, which in turn is in equilibrium with carbonic acid. The acid has a tendency to lose its hydrogen ions, forming bicarbonate ion, which it is also in equilibrium with. The bicarbonate can also lose a hydrogen ion, forming carbonate. Keep in mind that these steps work in reverse; in particular, carbonate ions can react with hydrogen ions to form bicarbonate.

When fossil fuels are burned, about a third of the resulting carbon dioxide ends up dissolved in the oceans. There, it undergoes a series of reactions, first combining with water to form carbonic acid, then losing protons one at a time. The result is a mixture of dissolved CO2, bicarbonate, and carbonate ions, as well as an increase in hydrogen ions, which increase the acidity of the seawater.

Burnette mentions a few principles which are useful for analyzing this scenario. The equilibration between gaseous CO2 and dissolved CO2 is expressed by Henry’s law, which states that the concentration of dissolved CO2 is proportional to the partial pressure of atmospheric CO2. This means that, as CO2 concentrations increase in the air, they will increase in the oceans too. Mathematically: