In attempting to arrive at a fair critique of Alexander de Seversky's "Combat Plane" concept, I discovered something unexpected about the "Combat Box" formation developed in late 1942 by then-Colonel Curtis LeMay. In particular, the numerical and spacial organization of the Combat Box involves the repetition of a single fundamental structure in three-dimensional space to build larger and more complex structures which, in turn, recapitulate the spacial form of the fundamental structure. In other words, Col. LeMay's geometric concept is fractal. Obviously, dire operational necessity in a lethal environment dictates that perfection in fractal geometrical form not be a priority, but just look at how close this formation comes to such perfection: Standard Group Combat Box Formation of 20 Aircraft - August 1943 from: http://www.303rdbg.com/formation.html Obviously, a more perfect fractal form would require a nine-plane squadron of three flights of three aircraft each, as part of a three-squadron group of 27 aircraft, with no Tail-End Charlies to protect the swallow-tail-shaped opening at the rear of the box from enemy fighters attempting to break up the formation from six o'clock level, but, once again, wartime necessity obviously militates against such theoretical perfection in favor of operational practicality. But the basics of fractal geometry are clearly present in Lemay's ideas, much as Pythagoras is wrongly credited with having discovered his eponymous theorem when he merely rediscovered what a group of tax-cheating Greek farmers had clearly figured out earlier.
I see nothing "fractal" about that but there certainly is some "self-similarity" which is one property of a fractal.
HallsofIvy: If self-similarity is a fractal property, then isn't the "Combat Box" somewhat fractal in nature (which is all I'm really claiming, as I've already freely admitted the lapses in self-similarity evident in LeMay's tactics, as he was a bomber pilot, and not a geometer)? All I'm claiming is that he discovered the essential gist of fractal geometry in pursuit of practical goals unrelated to geometrical theory, while (unbeknownst to him) significantly advancing geometrical theory.
Hello again, HallsofIvy: I've looked up contractions, and have found nothing in the equations which disproves my theory. Indeed, I've found this equation: dist((x, y), (x', y')) = ((x - x')2 + (y - y')2)1/2 which is clearly the Ancient Greek Tax Cheaters' Theorem (currently known under the name of the guy who RE-discovered it), of which Gen. LeMay could not possibly have been unaware. Could you please state the nature of your attack upon my idea more clearly, and in mathematical, rather than verbal, format?
The fact that something has one of the several required properties of a fractal does not mean it is fractal. That is all I am saying. I don't know what "contractions" you are talking about. I note that Wikipedia defines "fractal" as "a rough or fragmented geometric shape that can be split into parts, each of which is (at least approximately) a reduced-size copy of the whole,"[1] a property called self-similarity. " That is NOT the definition fractal I learned. I learned a fractal as a set the has fractional Hausdorf dimension. "Self similarity" can give fractional dimension but can also give integer dimension.
OK, so I've just read up on Hausdorff dimensions (most of which consists of stuff I knew about before I knew what girls were good for), and I have no problem with the concept that the "Combat Box" is fractal geometry at a level of simplicity sufficient to allow description either as fractal or integer. What I'm saying is that you still haen't proven that it's NOT fractal.
By the way, while looking up Hausdorff dimensions, I stumbled over the Sierpinski Triangle, whose dimensions log 3 / log 2 (approximately 1.585) seem, to me, to be, possibly, significantly near the Golden Section (1.618033988749895... ).
Anyways, here's the one-sheet that started it all: A Fair Critique of the de Seversky Combat Plane Alexander de Seversky was best known as a pioneer aviator and as the originator of the concept of the "Combat Plane", which is a strategic bomber with range, armor, and armament sufficient to enable deep=penetration bombing raids into enemy territory without need of fighter escort. While many consider this concept to have been foolhardy in the light of the experience hard-gained by air forces during the Second World War, I believe the experience of Eight United States Air Force over Central Europe during the last two years of the war validate the concept of the Combat Plane, however lacking in reality the overall concept of the ability of strategic aerial bombardment to eliminate an enemy's ability to wage war eventually proved to be. While it is true that most attempts to implement de Seversky's Combat Plane concept failed (largely due to the design of the aircraft intended to implement this idea (i.e., the Consolidated B-24 Liberator and the Avro Lancaster)), the combination of the Boeing B-17 Flying Fortress (with her famous ball turret emplaced so as to cover the plane's ventral aspect) with Curtis LeMay's "Combat Box" formation, designed to give every machine gun aboard every plane a clear field of fire into airspace beyond the box itself, as well as to facilitate a concentrated drop zone for the bombs and to keep the trailing planes free of the wake turbulence of the planes ahead of them, did, in fact, prove the validity of de Seversky's fundamental concept of the Combat Plane. The proof went to the extent that the US deep-penetration raids against Berlin in the spring of 1944 were, fundamentally, not actually intended to damage Berlin, but, rather, to attract German fighters which would be forced to defend the Capital, the specific purpose of this deliberate baiting being to clear the skies of German fighters in time for the invasion of Normandy. The German fighter force in the West was still formidable when the raids began, but virtually non-existent on D-Day, due to the effectiveness of the B-17 formed into the Combat Box as an anti-aircraft platform. The truth is that the B-17 accounted for more German aircraft than any other type in the ETO, fighters included.
How is [itex]\log_{2}{(3)} = \log{(3)}/\log{(2)}[/itex] "similar" to [itex](1 + \sqrt{5})/2[/itex]? And what does the golden ratio have to do with fractals?
I don't know. Maybe I'm wrong, or maybe there's cause for further research to find a non-obvious relationship.
HallsofIvy: In one of your replies, you implied that I get my information from Wikipedia. I somehow failed to note this statement, but I wish to ensure you that I obtain my information from reliable sources. This definition of fractals comes from the source I actually consulted, which is a page of the Yale University website (and, yes, I'm fully aware of the recent decline in Yale's academic standing, largely due to undergraduate students' parents' horror of the surrounding neighborhood, but it certainly is more reputable than Wikipedia): http://classes.yale.edu/fractals/ "Here we introduce some basic geometry of fractals, with emphasis on the Iterated Function System (IFS) formalism for generating fractals. In addition, we explore the application of IFS to detect patterns, and also several examples of architectural fractals. First, though, we review familiar symmetries of nature, preparing us for the new kind of symmetry that fractals exhibit. A. The geometric characterization of the simplest fractals is self-similarity: the shape is made of smaller copies of itself. The copies are similar to the whole: same shape but different size." *** I know that your assumption was the product of my failure properly to cite my sources, but I assure you that my knowledge of fractal geometry is fairly sophisticated, or I wouldn't have been able to recognize it when reading a page concerning Second World War strategic bombing formations.
But, the iterative self-similarity goes infinite numbers of times. This gives the essential fractal dimension of the object. In your case, the fundamental objects, airplanes, cannot be made arbitrarily small due to physical constraints. Therefore, one has to go to infinity in the opposite direction, namely expanding this structure without bound. For this you would need a formation with an infinite amount of planes, which, again, is physically impossible.
Every leaf on every tree is constructed according to the principles of fractal geometry, as is every tree, yet no leaf or tree is of infinite size. There is no more need for an infinite number of bombers than there is for infinitely large tree leaves. In theory, LeMay's concept can be scaled up to contain an infinite number of bombers. That's all that's necessary to constitute an IFS. The lack of an observed example of infinity in a real-world structure hardly constitutes a rational counter-argument to any claim of the fractal nature of such a structure. Do you think I'm an idiot?
This is irrelevant to the discussion. A finite object can have fractal dimensions if you go iteratively to infinity to smaller and smaller scales. No leaf has actually a fractal dimension. Leaves are not fractals, just as they are not triangles.
I appreciate your forthright honesty. I never claimed that leaves are fractals, or actually have fractal dimensions. Read what I actually wrote: I limited my claim to the statement that leaves are ORGANIZED ACCORDING TO FRACTAL PRINCIPLES. Just as I would say that religious people try their best not to sin, but no truly religious person would claim to be without sin: the sinfulness of their lives is, to them, a revelation of their dependency upon supernatural aid for the salvation of their souls, but it hardly proves that they're not religious.
Also, notice the year when Hausdorff introduced this concept: http://en.wikipedia.org/wiki/Hausdorff_dimension and compare it to the year stated in the title of your thread.
WHAT IS THIS, A DISSERTATION DEFENSE????? As the great Joey Ramone once said: "What is this? What's in it for me?" I've actually defined this concept in my last couple of posts. Nature, being economical, limits herself to a few structures which she repeats at different scales. Look at how the vascular systems of leaves emulate tree branches, and how tree branches emulate trees. Look at how atoms resemble solar systems and galaxies. Fractal geometry is a geometrical framework for understanding nature, the proof of which does not require any slavish emulation on the part of nature, which geometry is ABSTRACTED from nature, which nature, in turn, posesses finite resources with which to implement abstract patterns. WHEW!!!