Wednesday, October 21, 2009
Tuesday, October 20, 2009
Tuesday, October 13, 2009
Monday, October 12, 2009
brainspect






Single photon emission computed tomography (SPECT, or less commonly, SPET) is a nuclear medicine tomographic[1] imaging technique using gamma rays. It is very similar to conventional nuclear medicine planar imaging using a gamma camera. However, it is able to provide true 3D information. This information is typically presented as cross-sectional slices through the patient, but can be freely reformatted or manipulated as required.
A lung SPECT / CT fusion image
Contents
[hide]
* 1 Principles
* 2 Application
o 2.1 Myocardial perfusion imaging
o 2.2 Functional brain imaging
+ 2.2.1 POEP
* 3 Reconstruction
* 4 Further reading
* 5 Typical SPECT acquisition protocols
* 6 References
* 7 See also
* 8 External links
[edit] Principles
In the same way that a plain X-ray is a 2-dimensional (2-D) view of a 3-dimensional structure, the image obtained by a gamma camera is a 2-D view of 3-D distribution of a radionuclide.
SPECT imaging is performed by using a gamma camera to acquire multiple 2-D images (also called projections), from multiple angles. A computer is then used to apply a tomographic reconstruction algorithm to the multiple projections, yielding a 3-D dataset. This dataset may then be manipulated to show thin slices along any chosen axis of the body, similar to those obtained from other tomographic techniques, such as MRI, CT, and PET.
SPECT is similar to PET in its use of radioactive tracer material and detection of gamma rays. In contrast with PET, however, the tracer used in SPECT emits gamma radiation that is measured directly, whereas PET tracer emits positrons which annihilate with electrons up to a few millimeters away, causing two gamma photons to be emitted in opposite directions. A PET scanner detects these emissions "coincident" in time, which provides more radiation event localization information and thus higher resolution images than SPECT (which has about 1 cm resolution).[1] SPECT scans, however, are significantly less expensive than PET scans, in part because they are able to use longer-lived more easily-obtained radioisotopes than PET.[2]
Because SPECT acquisition is very similar to planar gamma camera imaging, the same radiopharmaceuticals may be used. If a patient is examined in another type of nuclear medicine scan but the images are non-diagnostic, it may be possible to proceed straight to SPECT by moving the patient to a SPECT instrument, or even by simply reconfiguring the camera for SPECT image acquisition while the patient remains on the table.
To acquire SPECT images, the gamma camera is rotated around the patient. Projections are acquired at defined points during the rotation, typically every 3–6 degrees. In most cases, a full 360 degree rotation is used to obtain an optimal reconstruction. The time taken to obtain each projection is also variable, but 15–20 seconds is typical. This gives a total scan time of 15–20 minutes.
Multi-headed gamma cameras can provide accelerated acquisition. For example, a dual headed camera can be used with heads spaced 180 degrees apart, allowing 2 projections to be acquired simultaneously, with each head requiring 180 degrees of rotation. Triple-head cameras with 120 degree spacing are also used.
Cardiac gated acquisitions are possible with SPECT, just as with planar imaging techniques such as MUGA. Triggered by Electrocardiogram (EKG) to obtain differential information about the heart in various parts of its cycle, gated myocardial SPECT can be used to obtain quantitative information about myocardial perfusion, thickness, and contractility of the myocardium during various parts of the cardiac cycle; and also to allow calculation of left ventricular ejection fraction, stroke volume, and cardiac output.
[edit] Application
SPECT can be used to complement any gamma imaging study, where a true 3D representation can be helpful. E.g. tumor imaging, infection (leukocyte) imaging, thyroid imaging or bone imaging.
Because SPECT permits accurate localisation in 3D space, it can be used to provide information about localised function in internal organs, such as functional cardiac or brain imaging.
[edit] Myocardial perfusion imaging
Main article: Myocardial perfusion imaging
Myocardial perfusion imaging (MPI) is a form of functional cardiac imaging, used for the diagnosis of ischemic heart disease. The underlying principle is that under conditions of stress, diseased myocardium receives less blood flow than normal myocardium. MPI is one of several types of cardiac stress test.
A cardiac specific radiopharmaceutical is administered. E.g. 99mTc-tetrofosmin (Myoview, GE healthcare), 99mTc-sestamibi (Cardiolite, Bristol-Myers Squibb). Following this, the heart rate is raised to induce myocardial stress, either by exercise or pharmacologically with adenosine, dobutamine or dipyridamole (aminophylline can be used to reverse the effects of dipyridamole).
SPECT imaging performed after stress reveals the distribution of the radiopharmaceutical, and therefore the relative blood flow to the different regions of the myocardium. Diagnosis is made by comparing stress images to a further set of images obtained at rest. As the radionuclide redistributes slowly, it is not usually possible to perform both sets of images on the same day, hence a second attendance is required 1-7 days later (although, with a Tl-201 myocardial perfusion study with dipyridamole, rest images can be acquired as little as two-hours post stress). However, if stress imaging is normal, it is unnecessary to perform rest imaging, as it too will be normal – thus stress imaging is normally performed
MPI has been demonstrated to have an overall accuracy of about 83% (sensitivity: 85%; specificity: 72%) [3], and is comparable with (or better than) other non-invasive tests for ischemic heart disease, including
[edit] Functional brain imaging
Usually the gamma-emitting tracer used in functional brain imaging is 99mTc-HMPAO (hexamethylpropylene amine oxime). 99mTc is a metastable nuclear isomer which emits gamma rays which can be detected by a gamma camera. When it is attached to HMPAO, this allows 99mTc to be taken up by brain tissue in a manner proportional to brain blood flow, in turn allowing brain blood flow to be assessed with the nuclear gamma camera.
[edit] POEP
Because blood flow in the brain is tightly coupled to local brain metabolism and energy use, the 99mTc-HMPAO tracer (as well as the similar 99mTc-EC tracer) is used to assess brain metabolism regionally, in an attempt to diagnose and differentiate the different causal pathologies of dementia. Meta analysis of many reported studies suggests that SPECT with this tracer is about 74% sensitive at diagnosing Alzheimer's disease vs. 81% sensitivity for clinical exam (mental testing, etc.). More recent studies have show accuracy of SPECT in Alzheimer diagnosis as high as 88%.[2] In meta analysis, SPECT was superior to clinical exam and clinical criteria (91% vs. 70%) in being able to differentiate Alzheimer's disease from vascular dementias.[3] This latter ability relates to SPECT's imaging of local metabolism of the brain, in which the patchy loss of cortical metabolism seen in multiple strokes differs clearly from the more even or "smooth" loss of non-occipital cortical brain function typical of Alzheimer's disease.
99mTc-HMPAO SPECT scanning competes with FDG PET scanning of the brain, which works to assess regional brain glucose metabolism, to provide very similar information about local brain damage from many processes. SPECT is more widely available, however, for the basic reason that the radioisotope generation technology is longer-lasting and far less expensive in SPECT, and the gamma scanning equipment is less expensive as well. The reason for this is that 99mTc (technetium-99m) is extracted from relatively simple technetium-99m generators, which are delivered to hospitals and scanning centers weekly to supply fresh radioisotope, whereas FDG PET relies on FDG which must be made in an expensive medical cyclotron and "hot-lab" (automated chemistry lab for radiopharmaceutical manufacture), then must be delivered directly to scanning sites, with delivery-fraction for each trip handicapped by its natural short 110-minute half-life.
[edit] Reconstruction
Reconstructed images typically have resolutions of 64×64 or 128×128 pixels, with the pixel sizes ranging from 3–6 mm. The number of projections acquired is chosen to be approximately equal to the width of the resulting images. In general, the resulting reconstructed images will be of lower resolution, have increased noise than planar images, and be susceptible to artifacts.
Scanning is time consuming, and it is essential that there is no patient movement during the scan time. Movement can cause significant degradation of the reconstructed images, although movement compensation reconstruction techniques can help with this. A highly uneven distribution of radiopharmaceutical also has the potential to cause artifacts. A very intense area of activity (e.g. the bladder) can cause extensive streaking of the images and obscure neighboring areas of activity. (This is a limitation of the filtered back projection reconstruction algorithm. Iterative reconstruction is an alternative algorithm which is growing in importance, as it is less sensitive to artifacts and can also correct for attenuation and depth dependent blurring).
Attenuation of the gamma rays within the patient can lead to significant underestimation of activity in deep tissues, compared to superficial tissues. Approximate correction is possible, based on relative position of the activity. However, optimal correction is obtained with measured attenuation values. Modern SPECT equipment is available with an integrated x-ray CT scanner. As X-ray CT images are an attenuation map of the tissues, this data can be incorporated into the SPECT reconstruction to correct for attenuation. It also provides a precisely registered CT image which can provide additional anatomical information.
[edit] Further reading
* Elhendy et al., Dobutamine Stress Myocardial Perfusion Imaging in Coronary Artery Disease, J Nucl Med 2002 43: 1634–1646
* For anyone interested in the brain-imaging applications of SPECT then this is a great review, although the full-text article is not available online without a subscription to the journal. The following link provides a link to the abstract only, and you might be able to access the full article through a membership with a medical library. W. Gordon Frankle, Mark Slifstein, Peter S. Talbot, and Marc Laruelle (2005). "Neuroreceptor Imaging in Psychiatry: Theory and Applications". International Review of Neurobiology, 67: 385–440.
[edit] Typical SPECT acquisition protocols
Study Radioisotope Emission energy (keV) Half-life Radiopharmaceutical Activity (MBq) Rotation (degrees) Projections Image resolution Time per projection (s)
Bone scan technetium-99m 140 6 hours Phosphonates / Bisphosphonates 800 360 120 128 x 128 30
Myocardial perfusion scan technetium-99m 140 6 hours tetrofosmin; Sestamibi 700 180 60 64 x 64 25
Brain scan technetium-99m 140 6 hours HMPAO; ECD 555-1110 360 64 128 x 128 30
Tumor scan iodine-123 159 13 hours MIBG 400 360 60 64 x 64 30
White cell scan indium-111 & technetium-99m 171 & 245 67 hours in vitro labelled leucocytes 18 360 60 64 x 64 30
[edit] References
1. ^ MeSH SPECT
2. ^ Bonte FJ, Harris TS, Hynan LS, Bigio EH, White CL (July 2006). "Tc-99m HMPAO SPECT in the differential diagnosis of the dementias with histopathologic confirmation". Clin Nucl Med 31 (7): 376–8. doi:10.1097/01.rlu.0000222736.81365.63. PMID 16785801.
3. ^ Dougall NJ, Bruggink S, Ebmeier KP (2004). "Systematic review of the diagnostic accuracy of 99mTc-HMPAO-SPECT in dementia". Am J Geriatr Psychiatry 12 (6): 554–70. doi:10.1176/appi.ajgp.12.6.554. PMID 15545324.
[edit] See also
* Gamma camera
* Neuroimaging
* Functional neuroimaging
* Magnetic resonance imaging
* Positron emission tomography
* ISAS (Ictal-Interictal SPECT Analysis by SPM)
[edit] External links
* Nuclear Medicine Information
* Mediso, GE HealthCare, Philips, Siemens – Major manufacturers of SPECT scanners
* ISAS (Ictal-Interictal SPECT Analysis by SPM) – Yale University
* The nuclear medicine and molecular medicine podcast – Podcast
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[show]
v • d • e
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2D: CXR · Angiography · Venography · AXR/KUB · DXA · Upper gastrointestinal series · Mammography · Dental radiography
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Retrieved from "http://en.wikipedia.org/wiki/Single_photon_emission_computed_tomography"
Categories: Nuclear medicine | Radiology | Radiobiology | Medical imaging | Neuroimaging | Medical physics
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Sunday, October 11, 2009
procedural coding
Procedural programming can sometimes be used as a synonym for imperative programming (specifying the steps the program must take to reach the desired state), but can also refer (as in this article) to a programming paradigm based upon the concept of the procedure call. Procedures, also known as routines, subroutines, methods, or functions (not to be confused with mathematical functions, but similar to those used in functional programming) simply contain a series of computational steps to be carried out. Any given procedure might be called at any point during a program's execution, including by other procedures or itself.[1] A procedural programming language provides a programmer a means to define precisely each step in the performance of a task. The programmer knows what is to be accomplished and provides through the language step-by-step instructions on how the task is to be done. Using a procedural language, the programmer specifies language statements to perform a sequence of algorithmic steps. Procedural programming is often a better choice than simple sequential or unstructured programming in many situations which involve moderate complexity or which require significant ease of maintainability. Possible benefits:
The ability to re-use the same code at different places in the program without copying it.
An easier way to keep track of program flow than a collection of "GOTO" or "JUMP" statements (which can turn a large, complicated program into spaghetti code).
The ability to be strongly modular or structured.
Some good examples of procedural programs are the Linux Kernel, GIT, Apache Server, and Quake III Arena.
Contents [hide]
1 Procedures and modularity
2 Comparison with imperative programming
3 Comparison with object-oriented programming
4 Comparison with logic programming
5 See also
6 References
7 External links
[edit]Procedures and modularity
Modularity is generally desirable, especially in large, complicated programs. Inputs are usually specified syntactically in the form of arguments and the outputs delivered as return values.
Scoping is another technique that helps keep procedures strongly modular. It prevents the procedure from accessing the variables of other procedures (and vice-versa), including previous instances of itself, without explicit authorization.
Less modular procedures, often used in small or quickly written programs, tend to interact with a large number of variables in the execution environment, which other procedures might also modify.
Because of the ability to specify a simple interface, to be self-contained, and to be reused, procedures are a convenient vehicle for making pieces of code written by different people or different groups, including through programming libraries.
(See Module and Software package.)
[edit]Comparison with imperative programming
Most procedural programming languages are also imperative languages, because they make explicit references to the state of the execution environment. This could be anything from variables (which may correspond to processor registers) to something like the position of the "turtle" in the Logo programming language.
[edit]Comparison with object-oriented programming
The focus of procedural programming is to break down a programming task into a collection of variables, data structures, and subroutines, whereas in object-oriented programming it is to break down a programming task into objects with each "object" encapsulating its own data and methods (subroutines). The most important distinction is whereas procedural programming uses procedures to operate on data structures, object-oriented programming bundles the two together so an "object" operates on its "own" data structure.
Nomenclature varies between the two, although they have much the same semantics:
object-oriented procedural
methods functions
objects modules
message argument
attribute variable
See Algorithms + Data Structures = Programs.
[edit]Comparison with logic programming
In logic programming, a program is a set of premises, and computation is performed by attempting to prove candidate theorems. From this point of view, logic programs are declarative, focusing on what the problem is, rather than on how to solve it.
However, the backward reasoning technique, implemented by SLD resolution, used to solve problems in logic programming languages such as Prolog, treats programs as goal-reduction procedures. Thus clauses of the form:
H :- B1, …, Bn.
have a dual interpretation, both as procedures
to show/solve H, show/solve B1 and … and Bn
and as logical implications:
B1 and … and Bn implies H.
Experienced logic programmers use the procedural interpretation to write programs that are effective and efficient, and they use the declarative interpretation to help ensure that programs are correct.
Saturday, October 10, 2009
Vide0 is liberated, v1deo as freedom
Trash Humpers by H. Korine
Rock my Religion by D. Graham (1o minute clip)
video i
Sierpinski Triangle & Fractiles
Sierpinski triangle (follow link for animated explanations)
The Sierpinski triangle (also with the original orthography Sierpiński), also called the Sierpinski gasket or the Sierpinski Sieve, is a fractal named after the Polish mathematician Wacław Sierpiński who described it in 1915.
Originally constructed as a curve, this is one of the basic examples of self-similar sets, i.e. it is a mathematically generated pattern that can be reproducible at any magnification or reduction.
Comparing the Sierpinski triangle or the Sierpinski carpet to equivalent repetitive tiling arrangements, it is evident that similar structures can be built into any rep-tile arrangements.
Contents [hide]
1 Construction
2 Properties
3 Analogues in higher dimensions
4 See also
5 References
6 External links
[edit]Construction
Animated construction. Click to enlarge.
An algorithm for obtaining arbitrarily close approximations to the Sierpinski triangle is as follows:
Note: each removed triangle (a trema) is topologically an open set.[1]
Start with any triangle in a plane (any closed, bounded region in the plane will actually work). The canonical Sierpinski triangle uses an equilateral triangle with a base parallel to the horizontal axis (first image).
Shrink the triangle to ½ height and ½ width, make three copies, and position the three shrunken triangles so that each triangle touches the two other triangles at a corner (image 2). Note the emergence of the central hole - because the three shrunken triangles can between them cover only 3/4 of the area of the original. (Holes are an important feature of Sierpinski's triangle.)
Repeat step 2 with each of the smaller triangles (image 3 and so on).
Note that this infinite process is not dependent upon the starting shape being a triangle—it is just clearer that way. The first few steps starting, for example, from a square also tend towards a Sierpinski triangle. Michael Barnsley used an image of a fish to illustrate this in his paper "V-variable fractals and superfractals."[2]
The actual fractal is what would be obtained after an infinite number of iterations. More formally, one describes it in terms of functions on closed sets of points. If we let da note the dilation by a factor of ½ about a point a, then the Sierpinski triangle with corners a, b, and c is the fixed set of the transformation da U db U dc.
This is an attractive fixed set, so that when the operation is applied to any other set repeatedly, the images converge on the Sierpinski triangle. This is what is happening with the triangle above, but any other set would suffice.
If one takes a point and applies each of the transformations da, db, and dc to it randomly, the resulting points will be dense in the Sierpinski triangle, so the following algorithm will again generate arbitrarily close approximations to it:
Start by labelling p1, p2 and p3 as the corners of the Sierpinski triangle, and a random point v1. Set vn+1 = ½ ( vn + prn ), where rn is a random number 1, 2 or 3. Draw the points v1 to v∞. If the first point v1 was a point on the Sierpiński triangle, then all the points vn lie on the Sierpinski triangle. If the first point v1 to lie within the perimeter of the triangle is not a point on the Sierpinski triangle, none of the points vn will lie on the Sierpinski triangle, however they will converge on the triangle. If v1 is outside the triangle, the only way vn will land on the actual triangle, is if vn is on what would be part of the triangle, if the triangle was infinitely large.
Animated creation of a Sierpinski triangle using the chaos game
Animated creation of a Sierpinski triangle using rule 90
Or more simply:
Take 3 points in a plane to form a triangle, you need not draw it.
Randomly select any point inside the triangle.
Move half the distance from that point to any of the 3 vertex points.
Plot the current position.
Repeat from step 3.
Note: This method is also called the Chaos game. You can start from any point outside or inside the triangle, and it would eventually form the Sierpinski Gasket with a few leftover points. It is interesting to do this with pencil and paper. A brief outline is formed after placing approximately one hundred points, and detail begins to appear after a few hundred.
Sierpinski triangle using IFS
Or using an Iterated function system
An alternative way of computing the Sierpiski triangle uses an Iterated function system and starts by a point at the origin (x0 = 0, y0 = 0). The new points are iteratively computed by randomly applying (with equal probability) one of the following three coordinate transformations (using the so called chaos game):
xn+1 = 0.5 xn
yn+1 = 0.5 yn; a half-size copy
This coordinate transformation is drawn in yellow in the figure.
xn+1 = 0.5 xn + 0.25
yn+1 = 0.5 yn + 0.5 ; a half-size copy shifted right and up
This coordinate transformation is drawn using red color in the figure.
xn+1 = 0.5 xn + 0.5
yn+1 = 0.5 yn; a half-size copy doubled shifted to the right
When this coordinate transformation is used, the triangle is drawn in blue.
Or using an L-system — The Sierpinski triangle drawn using an L-system.
bitwise XOR - The values of the discrete, 2D XOR function, z=XOR(x,y) also exhibit structures related to the Sierpinski triangle.
Other means — The Sierpinski triangle also appears in certain cellular automata (such as Rule 90), including those relating to Conway's Game of Life. The automaton "12/1" when applied to a single cell will generate four approximations of the Sierpinski triangle.
[edit]Properties
The Sierpinski triangle has Hausdorff dimension log(3)/log(2) ≈ 1.585, which follows from the fact that it is a union of three copies of itself, each scaled by a factor of 1/2.
If one takes Pascal's triangle with 2n rows and colors the even numbers white, and the odd numbers black, the result is an approximation to the Sierpinski triangle. More precisely, the limit as n approaches infinity of this parity-colored 2n-row Pascal triangle is the Sierpinski triangle.
The area of a Sierpinski triangle is zero (in Lebesgue measure). This can be seen from the infinite iteration, where we remove 25% of the area left at the previous iteration. Therefore the proportion of even numbers in Pascal's triangle must tend to 1 as the number of rows of the triangle tends to infinity.[citation needed]
[edit]Analogues in higher dimensions
A Sierpinski square-based pyramid and its 'inverse'
A Sierpiński triangle-based pyramid as seen from above (4 main sections highlighted). Note the self-similarity, so that this angle could be a 2D fractal in itself.
The tetrix is the three-dimensional analogue of the Sierpinski triangle, formed by repeatedly shrinking a regular tetrahedron to one half its original height, putting together four copies of this tetrahedron with corners touching, and then repeating the process. This can also be done with a square pyramid and five copies instead.[citation needed]
A tetrix constructed from an initial tetrahedron of side-length L has the property that the total surface area remains constant with each iteration.
The initial surface area of the (iteration-0) tetrahedron of side-length L is . At the next iteration, the side-length is halved
and there are 4 such smaller tetrahedra. Therefore, the total surface area after the first iteration is:
This remains the case after each iteration. Though the surface area of each subsequent tetrahedron is 1/4 that of the tetrahedron in the previous iteration, there are 4 times as many -- thus maintaining a constant total surface area.
The total enclosed volume, however, is geometrically decreasing (factor of 0.5) with each iteration and asymptotically approaches 0 as the number of iterations increases. In fact, it can be shown that, while having fixed area, it has no 3-dimensional character! The Hausdorff dimension of such a construction is which agrees with the finite area of the figure. (A Hausdorff dimension between 2 and 3 would indicate 0 volume and infinite area.)
The Sierpinski triangle (also with the original orthography Sierpiński), also called the Sierpinski gasket or the Sierpinski Sieve, is a fractal named after the Polish mathematician Wacław Sierpiński who described it in 1915.
Originally constructed as a curve, this is one of the basic examples of self-similar sets, i.e. it is a mathematically generated pattern that can be reproducible at any magnification or reduction.
Comparing the Sierpinski triangle or the Sierpinski carpet to equivalent repetitive tiling arrangements, it is evident that similar structures can be built into any rep-tile arrangements.
Contents [hide]
1 Construction
2 Properties
3 Analogues in higher dimensions
4 See also
5 References
6 External links
[edit]Construction
Animated construction. Click to enlarge.
An algorithm for obtaining arbitrarily close approximations to the Sierpinski triangle is as follows:
Note: each removed triangle (a trema) is topologically an open set.[1]
Start with any triangle in a plane (any closed, bounded region in the plane will actually work). The canonical Sierpinski triangle uses an equilateral triangle with a base parallel to the horizontal axis (first image).
Shrink the triangle to ½ height and ½ width, make three copies, and position the three shrunken triangles so that each triangle touches the two other triangles at a corner (image 2). Note the emergence of the central hole - because the three shrunken triangles can between them cover only 3/4 of the area of the original. (Holes are an important feature of Sierpinski's triangle.)
Repeat step 2 with each of the smaller triangles (image 3 and so on).
Note that this infinite process is not dependent upon the starting shape being a triangle—it is just clearer that way. The first few steps starting, for example, from a square also tend towards a Sierpinski triangle. Michael Barnsley used an image of a fish to illustrate this in his paper "V-variable fractals and superfractals."[2]
The actual fractal is what would be obtained after an infinite number of iterations. More formally, one describes it in terms of functions on closed sets of points. If we let da note the dilation by a factor of ½ about a point a, then the Sierpinski triangle with corners a, b, and c is the fixed set of the transformation da U db U dc.
This is an attractive fixed set, so that when the operation is applied to any other set repeatedly, the images converge on the Sierpinski triangle. This is what is happening with the triangle above, but any other set would suffice.
If one takes a point and applies each of the transformations da, db, and dc to it randomly, the resulting points will be dense in the Sierpinski triangle, so the following algorithm will again generate arbitrarily close approximations to it:
Start by labelling p1, p2 and p3 as the corners of the Sierpinski triangle, and a random point v1. Set vn+1 = ½ ( vn + prn ), where rn is a random number 1, 2 or 3. Draw the points v1 to v∞. If the first point v1 was a point on the Sierpiński triangle, then all the points vn lie on the Sierpinski triangle. If the first point v1 to lie within the perimeter of the triangle is not a point on the Sierpinski triangle, none of the points vn will lie on the Sierpinski triangle, however they will converge on the triangle. If v1 is outside the triangle, the only way vn will land on the actual triangle, is if vn is on what would be part of the triangle, if the triangle was infinitely large.
Animated creation of a Sierpinski triangle using the chaos game
Animated creation of a Sierpinski triangle using rule 90
Or more simply:
Take 3 points in a plane to form a triangle, you need not draw it.
Randomly select any point inside the triangle.
Move half the distance from that point to any of the 3 vertex points.
Plot the current position.
Repeat from step 3.
Note: This method is also called the Chaos game. You can start from any point outside or inside the triangle, and it would eventually form the Sierpinski Gasket with a few leftover points. It is interesting to do this with pencil and paper. A brief outline is formed after placing approximately one hundred points, and detail begins to appear after a few hundred.
Sierpinski triangle using IFS
Or using an Iterated function system
An alternative way of computing the Sierpiski triangle uses an Iterated function system and starts by a point at the origin (x0 = 0, y0 = 0). The new points are iteratively computed by randomly applying (with equal probability) one of the following three coordinate transformations (using the so called chaos game):
xn+1 = 0.5 xn
yn+1 = 0.5 yn; a half-size copy
This coordinate transformation is drawn in yellow in the figure.
xn+1 = 0.5 xn + 0.25
yn+1 = 0.5 yn + 0.5 ; a half-size copy shifted right and up
This coordinate transformation is drawn using red color in the figure.
xn+1 = 0.5 xn + 0.5
yn+1 = 0.5 yn; a half-size copy doubled shifted to the right
When this coordinate transformation is used, the triangle is drawn in blue.
Or using an L-system — The Sierpinski triangle drawn using an L-system.
bitwise XOR - The values of the discrete, 2D XOR function, z=XOR(x,y) also exhibit structures related to the Sierpinski triangle.
Other means — The Sierpinski triangle also appears in certain cellular automata (such as Rule 90), including those relating to Conway's Game of Life. The automaton "12/1" when applied to a single cell will generate four approximations of the Sierpinski triangle.
[edit]Properties
The Sierpinski triangle has Hausdorff dimension log(3)/log(2) ≈ 1.585, which follows from the fact that it is a union of three copies of itself, each scaled by a factor of 1/2.
If one takes Pascal's triangle with 2n rows and colors the even numbers white, and the odd numbers black, the result is an approximation to the Sierpinski triangle. More precisely, the limit as n approaches infinity of this parity-colored 2n-row Pascal triangle is the Sierpinski triangle.
The area of a Sierpinski triangle is zero (in Lebesgue measure). This can be seen from the infinite iteration, where we remove 25% of the area left at the previous iteration. Therefore the proportion of even numbers in Pascal's triangle must tend to 1 as the number of rows of the triangle tends to infinity.[citation needed]
[edit]Analogues in higher dimensions
A Sierpinski square-based pyramid and its 'inverse'
A Sierpiński triangle-based pyramid as seen from above (4 main sections highlighted). Note the self-similarity, so that this angle could be a 2D fractal in itself.
The tetrix is the three-dimensional analogue of the Sierpinski triangle, formed by repeatedly shrinking a regular tetrahedron to one half its original height, putting together four copies of this tetrahedron with corners touching, and then repeating the process. This can also be done with a square pyramid and five copies instead.[citation needed]
A tetrix constructed from an initial tetrahedron of side-length L has the property that the total surface area remains constant with each iteration.
The initial surface area of the (iteration-0) tetrahedron of side-length L is . At the next iteration, the side-length is halved
and there are 4 such smaller tetrahedra. Therefore, the total surface area after the first iteration is:
This remains the case after each iteration. Though the surface area of each subsequent tetrahedron is 1/4 that of the tetrahedron in the previous iteration, there are 4 times as many -- thus maintaining a constant total surface area.
The total enclosed volume, however, is geometrically decreasing (factor of 0.5) with each iteration and asymptotically approaches 0 as the number of iterations increases. In fact, it can be shown that, while having fixed area, it has no 3-dimensional character! The Hausdorff dimension of such a construction is which agrees with the finite area of the figure. (A Hausdorff dimension between 2 and 3 would indicate 0 volume and infinite area.)
Thursday, October 8, 2009
Tuesday, October 6, 2009
fuck yourself



Autogamy
Self-fertilization (also known as autogamy) occurs in hermaphroditic organisms where the two gametes fused in fertilization come from the same individual. They are bound and all the cells merge to form one new gamete.

fusion of male and female gametes (sex cells) produced by the same individual. Self-fertilization occurs in bisexual organisms, including most flowering plants, numerous protozoans, and many invertebrates. Autogamy, the production of gametes by the division of a single parent cell, is frequently found in unicellular organisms such as the protozoan Paramecium. These organisms, however, may also reproduce by means of conjugation, in which cross-fertilization is achieved by the exchange of genetic material across a cytoplasmic bridge between two individuals. Likewise, among higher plants, most of which are monoecious (i.e., bisexual—male and female gametes being produced by the same individual), most self-pollinating species are also capable of cross-fertilization, and even those that are obligate self-fertilizers are occasionally cross-pollinated by accident. Hermaphroditic animals (those in which both male and female gonads are borne on one individual) are rarely capable of self-fertilization, since many such species have adaptations encouraging cross-fertilization.
As an evolutionary and reproductive mechanism, self-fertilization allows an isolated individual to create a local population and stabilizes desirable genetic strains, but it fails to provide a significant degree of variability within a population and thereby limits the possibilities for adaptation to environmental change.
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