macro2009 Loc: Cheshire United Kingdom

A Romanesque Cauliflower

A Romanesque Cauliflower

Dan D Loc: The Valleys and Mountains of California

Wow! thanks.(wish I could download this one, and look at it closer!)

Erv Loc: Medina Ohio

Wow very well done!!!
Erv

rebride

Wow, too.
I probably would be still figuring out what it is if you hadn't said it was a cauliflower.

macro2009 Loc: Cheshire United Kingdom

Thanks Dan D, Erv and rebride funny vegetables I know.

This peculiar-looking vegetable is actually part of the cabbage family. Romanesque also goes by the name Romanesco broccoli, or Romanesque cabbage, but it doesn't look like any cabbage you've ever seen. The bright, florescent lime-green heads have tightly-pointed florets, and look like something from the dinosaur age.

Romanesque, which hails from Italy, has a crunchy, nutty taste, much different than cauliflower. Prepare Romanesque as you would cauliflower or broccoli. It is surprisingly sweet and mild.

A cup of Romanesque has about 25 calories and is a good source of potassium. It has vitamin A and slightly more vitamin C than white cauliflower.

How to Select: Look for tight heads that are heavy for their size, and a vibrant green color. Avoid ones with brown spots or small heads. Check the stem to make sure it's creamy white.

How to Store: Keep in the refrigerator for two to three days. After that it starts going soft and slightly browns.



Roasted Romanesque Cauliflower

One 2 1/2 pound Romanesque cauliflower, core removed and cauliflower broken up into florets
4 tablespoons extra virgin olive oil
1/4 teaspoon red pepper flakes
1/2 teaspoons coarse kosher salt
Several grinds black pepper

Preheat the oven to 425°F.

Place all ingredients in bowl mixing bowl. Toss and mix the Romanesque until evenly coated with seasoning and oil. Arrange florets in an even layer on a baking sheet.

Roast for 20 minutes, or until the Romanesque is tender. Finish with a squeeze of fresh lemon juice, if desired.

Backpacker Loc: Missouri

Nice! Never would have guessed that one.

carlysue Loc: Columbus

My kind of photographing! This is so very cool. There is such beauty in objects we don't see until we stop and take a closer look. Beautifully composed.

macro2009 Loc: Cheshire United Kingdom

Well thank you very much carlysue, I like taking images of food.

Dan D Loc: The Valleys and Mountains of California

Utterly fantastic, good images and good recipes. Thank you!

macro2009 Loc: Cheshire United Kingdom

Thanks Dan D I try to be a little selective with my images, and if background information is available so much the better; it helps with the experience!

Bret Loc: Dayton Ohio

Yes very nice shot.

macro2009 Loc: Cheshire United Kingdom

Thanks Bret very much appreciated.

CanonFire Loc: Phoenixville, PA

Interesting. The pattern looks very much like a Mandalbrot fractal set.

Shelley Loc: Lakeville, MN

I love the texture and the coloring is fantastic.

macro2009 Loc: Cheshire United Kingdom

Formal definition

The Mandelbrot set M is defined by a family of complex quadratic polynomials

P_c:\mathbb C\to\mathbb C

given by

P_c: z\mapsto z^2 + c,

where c is a complex parameter. For each c, one considers the behavior of the sequence

(0, P_c(0), P_c(P_c(0)), P_c(P_c(P_c(0))), \ldots)

obtained by iterating P_c(z) starting at critical point z = 0, which either escapes to infinity or stays within a disk of some finite radius. The Mandelbrot set is defined as the set of all points c such that the above sequence does not escape to infinity.
A mathematician's depiction of the Mandelbrot set M. A point c is coloured black if it belongs to the set, and white if not. Re[c] and Im[c] denote the real and imaginary parts of c, respectively.

More formally, if P_c^n(z) denotes the nth iterate of P_c(z) (i.e. P_c(z) composed with itself n times), the Mandelbrot set is the subset of the complex plane given by

M = \left\{c\in \mathbb C : \exists s\in \mathbb R, \forall n\in \mathbb N, |P_c^n(0)| \le s \right\}.

As explained below, it is in fact possible to simplify this definition by taking s=2.

Mathematically, the Mandelbrot set is just a set of complex numbers. A given complex number c either belongs to M or it does not. A picture of the Mandelbrot set can be made by colouring all the points c which belong to M black, and all other points white. The more colourful pictures usually seen are generated by colouring points not in the set according to how quickly or slowly the sequence |P_c^n(0)| diverges to infinity. See the section on computer drawings below for more details.

The Mandelbrot set can also be defined as the connectedness locus of the family of polynomials P_c(z). That is, it is the subset of the complex plane consisting of those parameters c for which the Julia set of P_c is connected.
Basic properties

The Mandelbrot set is a compact set, contained in the closed disk of radius 2 around the origin. In fact, a point c belongs to the Mandelbrot set if and only if

|P_c^n(0)|\leq 2 for all n\geq 0.

In other words, if the absolute value of P_c^n(0) ever becomes larger than 2, the sequence will escape to infinity.
Correspondence between the Mandelbrot set and the bifurcation diagram of the logistic map

The intersection of M with the real axis is precisely the interval [-2, 0.25]. The parameters along this interval can be put in one-to-one correspondence with those of the real logistic family,

z\mapsto \lambda z(1-z),\quad \lambda\in[1,4].\,

The correspondence is given by

c = \frac\lambda2\left(1-\frac\lambda2\right).

In fact, this gives a correspondence between the entire parameter space of the logistic family and that of the Mandelbrot set.

The area of the Mandelbrot set is estimated to be 1.50659177 ± 0.00000008.[13]

Douady and Hubbard have shown that the Mandelbrot set is connected. In fact, they constructed an explicit conformal isomorphism between the complement of the Mandelbrot set and the complement of the closed unit disk. Mandelbrot had originally conjectured that the Mandelbrot set is disconnected. This conjecture was based on computer pictures generated by programs which are unable to detect the thin filaments connecting different parts of M. Upon further experiments, he revised his conjecture, deciding that M should be connected.

The dynamical formula for the uniformisation of the complement of the Mandelbrot set, arising from Douady and Hubbard's proof of the connectedness of M, gives rise to external rays of the Mandelbrot set. These rays can be used to study the Mandelbrot set in combinatorial terms and form the backbone of the Yoccoz parapuzzle.[14]

The boundary of the Mandelbrot set is exactly the bifurcation locus of the quadratic family; that is, the set of parameters c for which the dynamics changes abruptly under small changes of c. It can be constructed as the limit set of a sequence of plane algebraic curves, the Mandelbrot curves, of the general type known as polynomial lemniscates. The Mandelbrot curves are defined by setting p0=z, pn=pn-12+z, and then interpreting the set of points |pn(z)|=2 in the complex plane as a curve in the real Cartesian plane of degree 2n+1 in x and y.
Other properties
Main cardioid and period bulbs
Periods of hyperbolic components

Upon looking at a picture of the Mandelbrot set, one immediately notices the large cardioid-shaped region in the center. This main cardioid is the region of parameters c for which P_c has an attracting fixed point. It consists of all parameters of the form

c = \frac\mu2\left(1-\frac\mu2\right)

for some \mu in the open unit disk.

To the left of the main cardioid, attached to it at the point c=-3/4, a circular-shaped bulb is visible. This bulb consists of those parameters c for which P_c has an attracting cycle of period 2. This set of parameters is an actual circle, namely that of radius 1/4 around -1.

There are infinitely many other bulbs tangent to the main cardioid: for every rational number \textstyle\frac{p}{q}, with p and q coprime, there is such a bulb that is tangent at the parameter

c_{\frac{p}{q}} = \frac{e^{2\pi i\frac pq}}2\left(1-\frac{e^{2\pi i\frac pq}}2\right).

Attracting cycle in 2/5-bulb plotted over Julia set (animation)

This bulb is called the \textstyle\frac{p}{q}-bulb of the Mandelbrot set. It consists of parameters which have an attracting cycle of period q and combinatorial rotation number \textstyle\frac{p}{q}. More precisely, the q periodic Fatou components containing the attracting cycle all touch at a common point (commonly called the \alpha-fixed point). If we label these components U_0,\dots,U_{q-1} in counterclockwise orientation, then P_c maps the component U_j to the component U_{j+p\,(\operatorname{mod} q)}.
Attracting cycles and Julia sets for parameters in the 1/2, 3/7, 2/5, 1/3, 1/4, and 1/5 bulbs
Cycle periods and antennae

The change of behavior occurring at c_{\frac{p}{q}} is known as a bifurcation: the attracting fixed point "collides" with a repelling period q-cycle. As we pass through the bifurcation parameter into the \textstyle\frac{p}{q}-bulb, the attracting fixed point turns into a repelling fixed point (the \alpha-fixed point), and the period q-cycle becomes attracting.
Hyperbolic components

All the bulbs we encountered in the previous section were interior components of the Mandelbrot set in which the maps P_c have an attracting periodic cycle. Such components are called hyperbolic components.

It is conjectured that these are the only interior regions of M. This problem, known as density of hyperbolicity, may be the most important open problem in the field of complex dynamics. Hypothetical non-hyperbolic components of the Mandelbrot set are often referred to as "queer" components.[citation needed]

For real quadratic polynomials, this question was answered positively in the 1990s independently by Lyubich and by Graczyk and Świątek. (Note that hyperbolic components intersecting the real axis correspond exactly to periodic windows in the Feigenbaum diagram. So this result states that such windows exist near every parameter in the diagram.)

Not every hyperbolic component can be reached by a sequence of direct bifurcations from the main cardioid of the Mandelbrot set. However, such a component can be reached by a sequence of direct bifurcations from the main cardioid of a little Mandelbrot copy (see below).

Each of the hyperbolic components has a centre, namely the point c such that the inner Fatou domain for P_c(z) has a super-attracting cycle (the attraction is infinite). This means that the cycle contains the critical point 0, so that 0 is iterated back to itself after some iterations. We therefore have that P_cn(0) = 0 for some n. If we call this polynomial Q^{n}(c) (letting it depend on c instead of z), we have that Q^{n+1}(c) = Q^{n}(c)^{2} + c and that the degree of Q^{n}(c) is 2^{n-1}. We can therefore construct the centres of the hyperbolic components, by successive solvation of the equations Q^{n}(c) = 0, n = 1, 2, 3, .... Note that for each step, we get just as many new centres as we have found so far.

I can see where you are coming from. Thanks for commenting.