Pineapples are herbaceous perennial bromeliads that produce sweet "multiple fruits", in that what we perceive as a single fleshy fruit body is actually a compression of adjacent, helically arranged fruits. The plant can grow to 5 feet tall, and is primarily pollinated by hummingbirds. In Hawaii, due to the extent of the agricultural cultivation of the plant, hummingbird importation is forbidden. The name "pineapple" was actually first recorded in 1398 to describe what we now call "pine cones". The term was redefined in 1664 after European explorers had discovered the tropical fruit, native to southern Brazil and Paraguay. We largely owe the modern distribution of pineapple to the Spanish, who spread it outside of Latin America (Columbus found it in the Caribbean) to Hawaii, the Philippines, Guam, and Zimbabwe. Pineapples are one of the relatively few plants that carry out crassulacean acid metabolism (CAM photosynthesis), whereby carbon dioxide is stored at night as the four-carbon acid malate and then released near the enzyme RuBisCO during the day, improving the efficiency of photosynthesis.

Monocots; Commelinids; Poales; Bromeliaceae; Ananas comosus

Pineapples contain large amounts of the protease bromelain, and as such raw pineapple should not be consumed by those suffering from various protein deficiencies, nor those with hemophilia, kidney, or liver diseases (or the occasional canker sore, as I'm well aware!). Other than that, the fruit is high in vitamin C and manganese, and helps with some intestinal disorders and may induce labor when a child is overdue. Due to the presence of the protease and the natural acidity of the fruit, pineapple makes a wonderful marinade for meats as a tenderizer, and is commonly used in pork dishes. It's sweetness, though, helps it get into many desserts, juices, smoothies, salads, etc. as well.

Basil is a low-growing annual herb of the mint (Lamiaceae) family, used frequently in Italian and Southeast Asian cuisine. The plant is native to South and Southwest Asia, having been cultivated there for over 5,000 years. Due to the abundance of phytochemicals in varying quantities, many cultivars exist with intrinsically different scents and flavors, depending on the chemical concentrations. The standard "sweet" or "Genovese" basil scent comes from eugenol, a phenylpropanoid compound also present in cloves, while lemon basil contains high amounts of citral, and licorice basil contains anethole. There are over 100 types of basil, each with its own unique fragrance.

Eudicots; Asterids; Lamiales; Lamiaceae; Ocimum basilicum

The word basil comes from the Greek word for "king", and was believed to have grown where St. Constantine and Helen discovered the biblical cross. Other religions around the world place importance on the plant, involving it with ritual and belief. The name is fitting, considering its designation as "the king of herbs" by many culinary experts. While many dishes use the herb as a flavoring additive, Italian pesto requires large amounts of the leaves blended with oil and pine nuts to create a thick sauce, and sometimes basil is used in fruit jams and jellies. And of course, as with many aromatic herbs, the plant has several health benefits, such as having antioxidant and anti-microbial properties.

On a cultivation side note, I grow a great deal of basil, and not always by choice; the plant, when exposed to the right amount of light, watering, and pruning, can grow VERY rapidly, often to the point of over-shading nearby herbs. I suppose this is not actually a problem, as it just means I need to make more pesto! Below is a recipe:

Italian Pesto

• 2 cups fresh Genovese (sweet) basil leaves
• 1/3 cup pine nuts
• 3 cloves garlic (I like garlic)
• 1/2 cup olive oil
• 1/2 cup Parmesan cheese

Pulse garlic with pine nuts in food processor. Add basil, and pulse again. While on, pour oil in steady stream into mixture, scraping down sides with spatula (don't let it hit the blades!). Add cheese and pulse until blended, with dash of salt and pepper.

After watching the recent South Park episode on the revised use of the word "faggot", I was reminded of how alive language is, and how neologisms and alteration of meaning can take place in a relatively short time frame. This got me thinking about the phrase "all right", which my uncle is always keen on reminding me is the accepted grammatically correct spelling. I admit that in formal writing, one should use the two word phrase. However, I do see a reason why the less common (but becoming more so) word "alright" should be adopted as a "real" word in its own right. And keep in mind that unlike most other languages of the world, English has no national or international academy or regulatory body that determines worthiness or correctness of words; there is only a loose confederation of dictionary publishers and a collection of traditionalists. Not one legal policy or official statute exists concerning purity of English (a far cry from French and Mandarin, whose language elite are, to put it bluntly, xenophobic and parochial).

The need for "alright" arises in the following context:

"Your answers to the problems were all right."

Here, there is some ambiguity over whether the speaker means that the answers were collectively correct (i.e., none of them were wrong), or that the solutions were satisfactory or mediocre. The difference can be ascribed to the fact that, in the former case, the two word phrase may be broken up: "All of your answers to the problems were right." In the latter case, "alright" should be used to imply that the answers were acceptable, though perhaps did not meet higher expectations, and cannot be recast as a split phrase. I feel that this difference, as well as the growing usage of the word in informal writing and dialogue, necessitates the acceptance of "alright" as a legitimate English word in the near future, if not already.

The plant grows as a small tree or woody vine and produces a fleshy berry with small edible seeds. The seeds (and indeed much of the fruit itself) can be bitter due to the presence of nicotinoid alkaloids, which are often found in Solanaceae (Nightshade family) plants like tomatoes, potatoes, peppers, and tobacco. The plant is native to India, and has appeared in the written record since 544 CE. Surprisingly, it was unknown to the Western world until about 1500 CE, and even then was not used widely due to concerns of the plant's toxicity, being a relative of nightshade. In British English the fruit is referred to as an "aubergine", whose origin can be traced through French, Catalan, Arabic, Persian, and Sanskrit. In the US, Canada, Australia, and New Zealand, the fruit is called an "eggplant" due to the appearance of some cultivars in 18th century England, which were yellow or white and resembled goose eggs. Today there are many varieties of different sizes, shapes, and colors, though the most common in the US are elongate ovoids with dark purple skin.

Eudicots; Asterids; Solanales; Solanaceae; Solanum melongena

The plant is used in cuisine around the globe, from Japan and India to Spain and Turkey. The fleshy part of the berry is able to absorb large amounts of fats and liquids, making it an ideal addition to sauces. Though the plant can be bitter, this can be alleviated by salting and rinsing the sliced fruit, and some cultivars are not bitter at all. The thin leathery skin is also edible, so peeling is not required. The plant could help with high blood pressure and free radical formation, and is a good source of folic acid and potassium. It has more nicotine than any other edible plant, though 20 pounds of eggplant would have to be eaten to match that in one cigarette.

Pomegranates are deciduous shrubs or small trees of the crape myrtle family, bearing grapefruit-sized berries with edible seeds and surrounding flesh (arils). A single fruit may contain up to 600 arils. The plant is native to Southwest Asia, and has been cultivated since ancient times, referenced often in Judaic and Greek mythos. In fact, dried pomegranates have been found in Egyptian tombs from the third millennium BCE, and the biblical city of Jericho. Today they are grown around the world wherever the climate is not too wet, due to their propensity for root fungus.

Eudicots; Rosids II; Myrtales; Lythraceae; Punica granatum

The word pomegranate comes from Latin, meaning "seeded apple", which has influenced the name in other languages as well. Our words "grenade" and "grenadine", as well as the Spanish city Grenada, owe their origin to the pomegranate, which was once used to make grenadine, and gave the city its name during the Moorish period. I assume the term grenade comes from the fruit's ability to blow its arils in every direction when thrown against a solid object. Judaism has had a long history of admiring the fruit, and associates it with righteousness as one of the seven foods special to the land of the Hebrews. In Greek mythology, Persephone was forced to stay in Hades four months out of the year, because she had been tricked into eating four pomegranate seeds. This explained the onset of winter, when her mother Demeter (goddess of the harvest) would mourn Persephone's return to the underworld. Pomegranates are also a common motif in Christian religious decoration as a symbol of Jesus' suffering and resurrection.

The fruit has many culinary uses, though the arils are often eaten raw. Pomegranate juice is a common ingredient in several Persian and Caucasian dishes and soups, or may be drunk on its own. When thickened and sweetened, it forms grenadine, included in many cocktails. The acidic tannins in the juice, along with the natural sugars, make it an excellent basis for glazes and sauces, applied to duck or other poultry. The fruit contains many beneficial phytochemicals, such as vitamin C, pantothenic acid, potassium, and antioxidant polyphenols. As such, it seems to inhibit many health problems, such as heart disease, high blood pressure, dental plaque, breast cancer, prostate cancer, diabetes, lymphoma and rhinovirus infection! For me, they are fun to eat, and the juice's citrus/berry flavor can't be beat.

Broccoli is a cultivar group of the cabbage family Brassicaceae, but amazingly is the same species as several other seemingly unrelated plants. Brassica oleracea also includes kale, collard greens, brussels sprouts, cabbage, kohlrabi and others, indicating that the species has undergone extensive cultivation from its wild form. The native variant is indigenous to limestone sea cliffs of Southern Europe, and is known to have first been cultivated about 2,000 years ago by the Roman Empire.

Eudicots; Rosids; Brassicales; Brassicaceae; Brassica oleracea

The part of the plant that we eat is the mass of flower buds that forms on the "head" (think cabbage). It may be boiled, steamed, or eaten raw, though boiling is discouraged to prevent loss of nutrients. It's high in vitamins A, C, and K, as well as dietary fiber. The vegetable also has several anti-viral, anti-bacterial, and anti-cancer properties as well. Being a cruciferous vegetable, the plant is potentially goitrogenic (can induce goiter formation), and should be thoroughly cooked before being eaten by people with thyroid problems or iodine deficiencies. Individuals sensitive to PTC (see cilantro entry) may find the flavor of Brassica members distasteful, but broccoli is one of my favorite vegetables. An excellent snack:

• 1 cup mayonnaise
• 1-2 tablespoons lemon juice
• 1 tablespoon curry powder
• raw broccoli

Mix the first three and dip broccoli. Quite tasty.

Now I'd like to examine physical theories that are more statistical in nature: statistical mechanics and quantum mechanics. Instead of assuming our system progresses exactly along the path of minimum action, we assign a probability distribution to weight the various paths it can take through phase space, with the most probable path being the one of stationary action. This formulation of statistical mechanics is much akin to Feynman's path integral approach to quantum mechanics, which I'll cover next time.

Let's start with a definition of the average action, , where is a probability distribution over all possible field configurations on the manifold of interest (spacetime), is our usual action functional along the path, and indicates that the integration is to be performed over all possible field configurations over all of the manifold. By definition, we must have . We may also define the entropy from Shannon's theory as . Employing the method of Lagrange multipliers to impose the constraints, we find that , where is the partition function for normalizing the probability, and is the Lagrange multiplier, with units of inverse action (in quantum mechanics, ). We also obtain , and . The most important result, though, comes from using Hamilton's Principle, . This means

But , so . Hence Hamilton's Principle applies to the mean action as well (which we would expect). This immediately yields for the expected path: the entropy change is maximized for the field configuration path we expect. This is essentially the second law of thermodynamics. In this sense, the second law and Hamilton's Principle are equivalent: a statistical process that extremizes action extremizes entropy change. Though the action is in terms of a Lagrangian, we can see that under certain circumstances, we may make a Legendre transform to put it in the form of a Hamiltonian. Additionally (the derivation is a bit too long), we can find that in that case the Lagrange multiplier is related to the temperature by defining , and then from the partition function we may derive the many laws and formulae of thermodynamics.

Perhaps the most beautiful and easy to understand field theory is that of electrodynamics. In the late 1800's, James Clerk Maxwell unified the rather disparate and seemingly unrelated topics of magnetism and electricity into a single theory, codified in the 4 equations below, Maxwell's equations:

The first equations is Gauss's Law, the second precludes the existence of magnetic monopoles, the third is Faraday's Law of induction, and the fourth is the Maxwell-Ampère law with a displacement current. These four partial differential equations can be modified slightly to be used in polarized, dielectric or magnetic materials, but this is essentially the entire theory of electromagnetism. To put these in a covariant framework (for relativistically invariant physics), we can introduce the rank 2 antisymmetric field tensor:

Using this and the definition , Maxwell's equations are and . In terms of the 4-potential , where , it simplifies even more: .

But how can we get this from a Lagrangian? Once again, we must construct some kind of scalar from the fields. From the field tensor, there is only one Lorentz invariant we may make: . (There is also a pseudo-scalar invariant related to the angle between the electric and magnetic fields, but the Lagrangian must be a true scalar). The only scalar we can form from the current is , so we choose as our Lagrangian , where the constants are chosen to conform to experiment. Invoking Hamilton's Principle by finding a stationary action,

Using the Euler-Lagrange equations exactly return the results above, so that this Lagrangian completely encapsulates electromagnetism. It is straightforward (if cumbersome) to generalize E&M to curved spacetime in the theory of general relativity.

First I'd like to apply Hamilton's Principle to Einstein's theory of General Relativity. We need to find a Lagrangian that incorporates the effect of mass on the manifold's metric. It must, as usual, be a scalar, and we suppose it is a functional of the metric and the matter fields. Hence, we suppose for free space

where is the invariant 4-volume element of our Riemannian manifold (and is the determinant of the metric), is a universal constant, and is the Ricci scalar (the simplest curvature invariant of a Riemannian manifold). If we were deriving the Einstein field equations for the first time, we would not know what these constants and scalars were, but we would still be able to write it in this form based on our assumptions of locality, isotropy of free space, etc.

Let us now suppose that the full action is this free-space action (the Einstein-Hilbert action) plus a term that describes matter fields:

When we apply the Euler-Lagrange equations from last time, we find that

We define the right hand side as ( times) the stress-energy tensor responsible for the curvature of spacetime due to the presence of matter. The left hand side is completely geometric (no physics is required), and from the theory of differential geometry, can be shown to equal , the Einstein tensor, in terms of the Ricci tensor, the Ricci scalar, and the metric. The Einstein field equations, then, are

This rather beautiful relation between geometry (left hand side) and physics (right hand side) can be viewed in either direction: a curving geometry tells matter and energy how to move; or equivalently, the presence of matter and energy curves spacetime. The theory does not, however, give us either one a priori, and the constant must be determined by experimental results. We can place restrictions on the stress-energy tensor, however: as the source of the gravitational field, we expect a particular simplicity in the non-relativistic limit (Newton's law of gravitation). Because of the symmetries of spacetime, we can apply Noether's theorem to obtain the conservation law . In many cases (where the spin tensor is zero), the tensor is symmetric and angular momentum is conserved as well. The radiation portions of the tensor can be gotten from Maxwell's theory, which I'll discuss next time. From the field equations, one can extract information about the nature of black holes, galactic rotation, etc., and as such these equations encapsulate most, if not all, of mechanics in the non-quantum regime.

Of late I have been intrigued by the idea that all differential equations that make up the bulk of our understanding of physics can be recast as integral equations, subject to some stationary condition. In fact, these all essentially reduce to a condition called Hamilton's Principle, stating that as a system evolves, the "action" is stationary. I will define these ideas in a bit. As I researched the subject, I found that not only classical mechanics could be formulated this way, but also optics, special relativity, general relativity, electromagnetism, quantum mechanics, and even statistical mechanics, essentially all branches of physics. Even string theory can be started from an action principle. The implications are amazing: essentially all of physics can be derived from one principle.

I suppose that now I should formalize: suppose we have an -dimensional manifold and a target manifold . Let be the set of smooth functions . Now consider a functional , where is a field. Now we must make some assumptions. First, by one postulate of quantum mechanics, the action must map to the field of real numbers , since observables (such as action) have real eigenvalues. Also, from relativity, we assume locality, as required for causality. Quantum entanglement does not pose a problem because information cannot be transmitted nonlocally instantaneously. Locality implies that if , we can assume depends only on a function of and its derivatives over the manifold . Hence, :

The function is called the "Lagrangian" function. The Euler-Lagrange equations discussed below can be modified to include higher order derivatives of the Lagrangian*, but through a substitution, the Euler-Lagrange equations can always be reduced to derivatives with respect to the function and its first derivative, nothing higher (at the expense of additional equations to solve). The Euler-Lagrange equations are derived using the calculus of variations, based on the functional derivative. We can imagine varying the function until the action integral is maximal or minimal or inflected (ie, "stationary"). Finding this stationary action is the essence of the functional derivative: normally we find a stationary point by setting the derivative of a function with respect to a variable to zero. Here we set the derivative of a functional with respect to a function to zero. There is an issue, in the derivation, of boundary conditions, so that we must specify on if is compact, or place some limit on it as . This gives us the Euler-Lagrange equations:

where for each of the dimensions of the manifold. So what is ? These are the physical fields of interest. In classical Lagrangian mechanics, they are the coordinates themselves, expressed as functions of time. In field theory, they are physical fields as functions of spacetime. Hence the target manifold is the set of field values at a given point. For example, in classical mechanics, we might have , so that

Hence, with a suitable choice of Lagrangian, all the laws of physics may be derived (though of course in practice this is definitely not the way to always proceed). The question that naturally arises is, "What is the total Lagrangian?" Well, it must be a scalar to preserve isotropy of space and homogeneity of spacetime (that is, empty space looks the same in all directions, and there is no difference between one point and the next). Aside from this, it may be chosen to describe the physics involved. Over a few more posts, I'll derive some important equations in physics from Hamilton's Principle.

*The Euler-Lagrange equations for a Lagrangian dependent on the first derivatives of are:

Well, I've completed the qualifying exam for physics (a two-day, 4-hour-each nightmare), and if I passed, I'll get my Master's degree. Besides the test, this weekend has been rather packed: Peter and Rika came down and spent a few days with us, Rachael came down, and Phil's parents visited. We played many games, celebrated Philip's 23rd birthday, saw "9", went to the Wild Animal Park, cooked a lot of food, watched Batman Begins and The Dark Knight...quite exhausting but enjoyable. Now it's time to continue with my research, and start classes on Thursday: non-equilibrium statistical physics, field theory, and solid state physics. It should be a good quarter. This weekend I'll be visiting the family and going to Catalina, and the next weekend Rachael will come down again for a whole week! And I might even have a car by then...

Apples are an example of what botanists call a pome: an accessory fruit produced by members of the subfamily Maloideae of the rose family. A pome's exocarp and mesocarp are formed from the carpels and make up the fleshy part of the fruit, while the endocarp forms a leathery or stony case around the seeds, commonly called the core. The end of the fruit opposite the stem is the calyx, where one can often see the remains of the flower's sepals, style, and stamens. The trees themselves are deciduous, between 10 and 40 feet tall, and produces five-petaled white and pink flowers in the spring. First being noticed in the wild at least 2500 years ago in Central Asia, there are now over 7,500 different cultivars, created over centuries of selection. Indeed, the apple tree may be the earliest domesticated tree. Part of this diversity may be due to the extreme heterozygous nature of the plants: apples grown from seed may be radically different from their parents, so that most varieties today are grown from grafting.

Eudicots; Rosales; Rosaceae; Maloideae; Malus domestica

Perhaps due to its long history, the apple has had an interesting impact on human culture. Many pagan religions make reference to apples as mystical or forbidden fruit, though the term "apple" was used to describe all fruits as late as the 17th century. The Latin word for apple, malus, is similar to that for evil: malum. This may provide the Christian myth of the "forbidden, evil apple" in the Garden of Eden, and how a man's "Adam's apple" comes from the fruit sticking in Adam's throat.

Short of choking to death, apples actually provide numerous health benefits, to the point that "An apple a day keeps the doctor away." It has been shown to decrease risks of lung, colon, and prostate cancer, as well decline in mental faculties. It can also help with weight loss and cholesterol levels. The seeds, however, are mildly poisonous, and should not be consumed in bulk. Their bitter flavor comes from amygdalin, the same cyanide compound found in bitter nuts. The fruits can be used in a wide number of culinary applications, from its juice (and ciders, sprits, and vinegars) to apple cakes, pies, crumbles, crisps, butters, jellies and sauces. They can be served spiced, caramelized, in salads, or just eaten raw (my favorite). So indulge in this "sinful(?)" fruit and enjoy it.

A grape is a true berry of the woody vines of genus Vitus. The vines are lianas: long stemmed perennial or deciduous woody vines that root in soil and use trees or other vertical support to reach the sunlit canopy, and as such are epiphytes (not parasitic). Most of today's cultivars were domesticated from the Common Grape Vine (Vitus vinifera), whose wild range extends across Europe, North Africa, and the Middle East. The wild variety (subspecies sylvestris) is dioecious, requiring pollination to fruit, but domesticated cultivars have hermaphrodite flowers. The genus has been dated to have evolved roughly 130 to 200 million years ago, and humans have interacted with it since the Neolithic Period began nearly 12,000 years ago.

Eudicots; Vitales; Vitaceae; Vitus vinifera

The plant was likely domesticated around 3500 BCE in Southwest Asia, and spread quickly to Egypt and Phoenicia. Grapes and wine production are mentioned in the Epic of Gilgamesh and many Egyptian hieroglyphics of the third millennium BCE, and the Greeks introduced wine to Europe in the Minoan Age (Hesiod and Homer make extensive reference to wine, and classical pantheons of gods usually include one deity devoted to the beverage, Dionysus or Bacchus). The Etruscans developed a trade network for the fruit and wine that extended beyond the Mediterranean, and the Chinese began its cultivation in the Han Dynasty, around the second century CE. The vine was introduced to the Americas with the first European explorers. Today, most grapes are produced by Italy, France, China, and the USA, with 98% of grapes in America grown (not surprisingly) in California. Why do we have a budget crisis in this state when we grow essentially all of the food in the country?

Grapes have a wide variety of health benefits, though most of them come from the now-uncommon seeded varieties, due to the more complex mixture of phytochemicals produced in seed development. Polyphenol antioxidants found in grape skins (like resveratrol) have been linked to reduction in cardiovascular problems, to the point where a glass of red wine per day is recommended (two for men). These antioxidants in addition to anthocyanins and other phenolics found in grape skins also limit gene expression of heart and skeletal aging, and so help prevent certain age-related health problems. In addition to fermenting to form wine (whose alcohol content has additional health benefits if consumed moderately), grapes can be eaten raw, mashed into jams and jellies, squeezed into juice, and dried into raisins and Zante currants. Grape seed oil and extracts are also used in many cosmetic and cooking applications, and even the leaves can be used in cooking and medicinal application.

In response to Phil's latest post, I think I'd like to clarify why the values are so agreeable between semiclassical Bohr theory, and relativistic Bohr theory. Let's start by deriving the allowed Bohr orbital radii in the relativistic correction. Assuming circular orbits, we balance the centripetal force with the (Lorentz invariant) Coulomb force:

where here I'm using Gaussian units for notational ease, is the mass of an electron, and . We also make the Bohr assumption that angular momentum is quantized in units of Dirac's constant:

Eliminating the radius from the system of equations gives us , where is the fine structure constant. Solving for the radius and using the definition of gamma, we find:

Where pm is the Bohr radius. In particular, the ground state is , about 99.997% of the Bohr radius. So we see that the relativistic correction makes the ground state radius slightly smaller, a factor easily lost in rounding of values. This result also gives an interesting insight into atomic structure: consider a hydrogenic (one electron) atom, of atomic number Z. Then everywhere in our calculations, we have . This is of particular interest in our expression for the velocity of the electron in the ground state:

Note that if Z is large enough (greater than 137), the velocity of the electron would exceed the speed of light. This puts an intrinsic limit on the size of atoms, namely that atomic numbers above 137 are essentially singular. Looking at the periodic table, we see that we have identified elements up to 118, so the results are in agreement.

In my previous post, I described spheres as being the most symmetric of all objects. Well, a fractal is practically the opposite, though still with a degree of cohesiveness. Loosely speaking, a fractal is a set of points in space that is infinitely self similar, created through a non-terminating recursive iteration. In other words, if you "zoom in" on any section of a fractal, you see a similar pattern as the fractal on the whole. For example, broccoli has this property (to a point): each little clipping of broccoli looks like a miniature head of broccoli. Clouds, lightening bolts, coastlines and snowflakes have this property as well. It should be noted, however, that fractals also have a Hausdorff dimension that is greater than their topological dimension, ie, they are really jagged and convoluted. A Euclidean line is self-similar (you zoom in on a line and it looks like a line), but its geometry is entirely representable in Euclidean terms (both the Hausdorff and topological dimensions are 1). Fractals usually have fractional dimension. For example, the Sierpinski carpet shown below, created by removing the middle ninth of a square, then removing the middle ninth of each of the surrounding eight squares, and so on, has a dimension of 1.8928.

It's almost 2 (an area), but a bit lower (towards 1, a line). Even though the square started as a 2 dimensional area, an infinite number of removals actually lowers its dimensionality. Another example is the Koch snowflake, below, with a dimensionality of 1.26. Thus it's mostly a 1-D curve, but is "dense" enough that it almost fills a bit of area. On zooming in on one corner, we see the same structure repeated indefinitely.

Real objects, of course, are not true fractals, in that if you zoom in enough the self-similarity breaks down, and we notice that a broccoli cell does not look like a head of broccoli. Fractal behavior can come into play when considering the real-world phase spaces of chaotic systems. Chaotic systems are inherently fractal, in that there is no long-term cyclic behavior, so that paths in the phase space of a chaotic particle never cross, but also never return to the same path or approach a single point/curve. As such, the paths "fill" regions of phase space, in some cases densely, so that it really is 2 dimensional, and in some cases more like a line, yielding a fractal dimension. Graphical representations of certain iterative functions can yield striking patterns like the Mandelbrot and Julia sets, again possessing detail at arbitrary scales.

Spheres truly are amazing mathematical objects. I would consider them the most symmetrical of all objects, in any dimension (the ball, circle, and interval being other variations). There are problems, however, with incorporating spheres into our rigid Cartesian way of thinking, where only right angles are appreciated. This leads to such problems as the Mercator projection for maps, causing some young children to believe that Greenland is the size of Africa. It is an unfortunate fact that we cannot isometrically map the surface of a sphere to a plane, due to the very different nature of the two objects (most importantly their differing curvatures). Even in our use of spherical coordinates, we must introduce problems: the poles are ill-defined. I'm not just talking about the periodicity that comes into account any time we include an angular variable; I mean that an infinitude of values are possible for the azimuthal angle at the poles. As such, we see that there exist singularities of a sort at the north and south poles of a sphere in spherical coordinates. And singularities are always interesting.

I was reminded today of one method of mapping the sphere to the plane I came across at Berkeley: stereographic projection. The idea is illustrated below:

A sphere (usually unit radius for ease) is placed on a plane, so that its south pole and the origin of the plane coincide. A line from the north pole is drawn to any point on the plane, and we identify the point it intersects the sphere with the point on the plane (P with P'). This eliminates any problems with the south pole, but to what point is the north pole mapped? Once again, this one point is mapped to an infinite number of "ideal points" at infinity on the plane. What I find amazing is the way to fix that: compactify the sphere and the plane. This means we make one more point, called "infinity", that corresponds to the "edge" of the plane and the north pole of the sphere. In so doing, we create an conformal (angles are preserved) isomorphism between the sphere and the compactified plane (actually a homeomorphism). In this way, any problem that we need to do on the plane, like an integral or finding if lines are parallel or intersection properties, can be transformed to one on the sphere, which is often easier. For instance, all lines AND circles on the plane are just circles on the sphere (straight lines pass through the north pole). Parallel lines on the plane intersect at the north pole on the sphere. Integrals that cover the whole real axis are just over a circle on the sphere, etc. I think the mapping is neat because it relates in a very clear way two very different representations of the same object (the compactified plane).

The kiwifruit, or kiwi, is native to southern China, but has primarily been exported by New Zealand since the 1950's. In fact, the name itself comes from the berry's similarity to the kiwi bird of New Zealand. The Chinese have adopted the kiwi as the national fruit of China in an effort to remind the public of the fruit's origin. The plant is a woody vine of the order Ericales, a rather diverse group of asterids including blueberries, tea, pitcher plants, persimmons, and azaleas. Common cultivars produce an oval berry, with fibrous brown skin and bright green flesh, with rows of small edible black seeds. The flavor is sweet and tart, rather unique, and almost citrus-like.

Eudicots; Asterids; Ericales; Actinidiaceae; Actinidia deliciosa

The fruit is rich in vitamin C and potassium, and serves as a natural blood thinner. The skin is very high in dietary fiber, and as such the fruit can be a mild laxative, though the skin is usually not consumed. It is also rich in the enzyme actinidin (same family of thiol proteases as papain), to which some people can have allergic reactions. The fruit also conatins calcium oxalate (read my starfruit entry). Thus the raw fruit should not be used in dishes containing milk or dairy products, or gelatin, as the enzymes dissolve collagen and casein. I found this out the hard way during a high school biology project, in which we attempted to make a food model of the cell. A hollowed watermelon served as the cell wall, and whole kiwis were used as chloroplasts. The cytosol, unfortunately, was chosen to be lime jello, and the kiwi proteases dissolved the jello overnight, resulting in a soup of sugary, sticky organelles the next day. I could never hold it against the kiwis, though...they are far too delicious, as the species name implies.

Perhaps a bit late for a "Monday" posting, but I've been busy. The starfruit, or carambola, comes from a slow-growing, short-trunked bushy tree growing to 20-30 feet. Its origin is thought to be Sri Lanka and islands of the Indian Ocean, but it has been mainly cultivated in Southeast Asia for centuries. The tree is tropical/subtropical, so introduced varieties have been grown in Hawaii, Florida, the Caribbean, and the South Pacific. The fruit is oblong, with 5-angled ridges running the length. It possesses a thin, waxy skin, and is yellow-orange in color. The entire fruit is edible, and the pericarp (flesh) is juicy, crisp, and yellow when fully ripe. They may contain up to 12 flat brown seeds.

Eudicots; Rosids I; Oxalidales; Oxalidaceae; Averrhoa carambola

The flavor is very complex, described as a combination of pineapple, orange, papaya, and grapefruit. Like grapefruit, starfruit contain rather significant quantities of oxalic acid, and in fact have the highest concentration of any edible plant. As such, people with kidney problems should avoid eating the fruit, as calcium oxalate is the primary constituent of kidney stones. Having the fruit with dairy products is also not recommended, as the oxalic acid easily denatures casein, forming a gritty precipitate of calcium oxalate. It should also be noted that starfruit is an inhibitor of many cytochrome P450 isoforms, and hence may interfere with certain medications. If you would not be adversely affected by ingesting the fruit, then I recommend trying it. Though it can be eaten out-of-hand, it is also commonly prepared in salads, as a garnish, in puddings, tarts, stews, curries, jams, jellies, syrups, relish, and dried. Carambola juice is used as a cooling, refreshing beverage as well. Below is a good recipe for pork with star fruit:

Star Fruit Pork Scaloppini

• 1 Tbsp butter
• 4 pork cutlets, 4 ounces each
• salt and pepper to taste
• 1 star fruit, sliced
• 1/2 cup beef broth
• 1/2 cup orange juice
• 1 cup whipping cream
• 2 Tbsp fresh coriander, chopped
• 2 tsp orange peel

Preheat oven to 200F. Melt butter in frying pan over medium heat. Cook pork about 4 to 5 minutes per side. Season with salt and pepper. Transfer cutlets to a serving dish and set aside in oven. In same pan, cook star fruit 1 minute per side. Set aside in oven. Add beef broth, orange juice and cream to saucepan; cook until reduced by half. Add coriander and orange peel. Continue cooking 2 to 3 minutes. Spoon sauce over meat and garnish with star fruit. Serve with rice and zucchini, if desired. Serves 4.

I recently needed to do a rather nasty integral involving arcsech, and though I was able to find it on the web/integrator/Mathematica/etc., I wondered where I might find a big poster of integrals that might be printed out and used for reference. To my astonishment, it seems like no such table exists for sale anywhere on the web, despite its obvious usefulness. I took it upon myself to create a table of integrals in poster format, which you can find here: http://www.blog.quantum-immortal.net/imgs/integrals.pdf in pdf format. For those that are interested, could you look it over and suggest any additional ones that should be included, or unnecessary ones that can be removed? Comments on layout/style are appreciated, but note that it is not in its finalized form. If you'd like a copy of the finished version, let me know and I can send you the final pdf. If you are a student or teacher, most likely your institution has a plotter capable of printing A0-sized poster formats for a rather small fee (<$5). Otherwise, you'd have to use Kinko's or a similar printing company to plot it, and it could be rather expensive. If the professional look isn't important, you could always use page tiling in Acrobat to print out many A4 pages and tape them together. Thanks guys.

Avocados are Magnoliids belonging to the Laurel family, Lauraceae. The leaves are like those of most Lauraceaens, simple and lanceolate (like Bay leaves), and the plant grows as a medium to tall tree (up to 70 ft). The plants are most well known for their fruit, a leathery-skinned berry with a pasty edible pericarp and hardened pit. The fruit is a true berry, and not a drupe, as the entire ovary wall ripens into the edible pericarp, whereas in a drupe, the ovary wall forms the outermost layer of the pit. The genus Persea has its origin in West Africa during the Paleocene, from whence it spread to Asia, South America, Europe, and North America. As Africa, western Asia, and the Mediterranean dried out in the middle of the Tertiary Period, and after the glaciation of Europe and Northern America in the Pleistocene, the genus went extinct in those regions, resulting in its current distribution. It is thought that avocados may be evolutionary "ghosts", in that the fruit seems adapted for now-extinct Pleistocene megafauna like Giant Ground Sloths or Gomphotheres. The large mammals would swallow the fruit whole and excrete the toxic pit in their dung, ready for sprouting. No extant animals are large enough to do this, and it's thought that avocados would have gone extinct or evolved a different fruit morphology were it not for human cultivation.

Magnoliidae; Laurales; Lauraceae; Persea americana

As for human cultivation, the plant has been grown in Central America since at least 900 CE, and likely much earlier. The name is derived from the Nahuatl word ahuacatl, meaning testicle, referring to the shape of the fruit. The Nahuatl word ahuacamolli, meaning "avocado soup", is the source of the Mexican Spanish word "guacamole". The fruits have had a long-standing stigma as sexual stimulants, and were known by the Aztecs as "the fertility fruit". Europeans at first avoided them to maintain the image of chastity, but by the mid-1500's, it became a popular crop. In the 19th century, it was introduced from Mexico into California, and avocado cultivation became a very successful business. 95% of US avocados come from California, with 60% from San Diego county, the most common cultivar being the Hass variety. It is interesting to note that as tasty and healthy they can be for humans (rich in vitamins B, E, and K, have more potassium than bananas, highest fiber content of any fruit, contain many healthy oils and fats), the fruit can be irritating or deadly to many other animals. There is evidence that cats, dogs, cattle, goats, rabbits, rats, birds, fish and horses can be harmed or killed by eating parts of the avocado plant. The toxicity is derived from persin, a poisonous fatty acid that has been recently isolated as a fungicide. It has also been shown to kill breast cancer cells. It seems like avocados are really useful for humans, and really bad for basically everything else.