Figure 3 Traditional geometry and spatial fractals

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Presentation transcript:

Figure 3 Traditional geometry and spatial fractals Figure 3 | Traditional geometry and spatial fractals. a | The regular Euclidean dimension, meaning the familiar geometrical descriptors, assigns an integer to each point or set of points in space: 0 to a point, 1 to a straight line, 2 to a plane surface, and 3 to a volume or 3D figure. Complex macroanatomical or microanatomical structures cannot be analysed by regular Euclidean geometry, but can be described quantitatively by fractal geometry as a fractal dimension (FD) falling inclusively between these integer topological dimensions. b | The FD for the trabeculae in a 2D digitized image such as this cardiovascular magnetic resonance cine left ventricular short-axis slice will lie inclusively between 1 and 2, and for a 3D image, between 2 and 3. c | The box-counting method works by applying a grid of boxes of side length (e) over the image of the fractal (>60 grids are needed for this contour, but only three are shown) and counting the number (Ne) of the smallest number of boxes of side length (e) required to cover the surface or outline of the object completely. d | The empirical box-counting FD (1.401 in this example) is estimated from the slope of a regression line when log(Ne) is plotted against log(1/e); therefore, FD = log (Ne)/log(1/e). e | Summary of the clinically relevant ranges of FD in health and disease derived from cardiovascular magnetic resonance research studies in the literature to date. All data reported as mean ± SD or mean and interquartile ranges (denoted by *). HCM, hypertrophic cardiomyopathy; HFpEF, heart failure with preserved ejection fraction; HFrEF, heart failure with reduced ejection fraction; LVNC, left ventricular noncompaction. Captur, G. et al. (2016) The fractal heart — embracing mathematics in the cardiology clinic Nat. Rev. Cardiol. doi:10.1038/nrcardio.2016.161