Some reflections on the emergence of spacefilling curves. In order for a nodelinkdiagram to be effective, the nodes must be arranged well on the screen. That is, the multidimensional index takes the ndimensional key spaces and maps them onto a onedimensional space and then the onedimensional retrieval system is activated. Download spacefilling curves universitext pdf by hans. We will show in this chapter that it is still possible for such curves to have a positive twodimensional lebesgue measure. Hans sagan books list of books by author hans sagan. The function visits double and triple points predicted by nettos theorem. Isbn 9780387942650 this book at amazon print version 9781461208716 this book at amazon electronic version doi ebook. Hans sagan, author of introduction to the calculus of variations, on librarything. Pushing lebesgue measure with peano curve mathematics. In mathematical analysis, a spacefilling curve is a curve whose range contains the entire. The idea that a onedimensional curve may completely cover an area or a volume was, at that time, completely novel and counterintuitive.
In this paper we propose to describe space filling and related curves by means of simultaneous nonlinear grammars. Hans sagans research works university of vienna, vienna. Jan 16, 2016 a montage of space filling curves, meant as a supplement to the hilbert curve video. Author of introduction to the calculus of variations, integral and differential calculus, boundary and eigenvalue problems in mathematical physics, beat the odds, calculus accompanied on the apple, introduction to the calculus of variations, boundary and eigenvalue problems in mathematical physics, advanced calculus. If you want to use this page offline, you must also download and unzip diagram. Contents preface vii acknowledgments ix illustration credits xi chapter 1. More editions of space filling curves universitext. A montage of space filling curves, meant as a supplement to the hilbert curve video.
Unfortunately, no comprehensive treatment has ever been attempted other than the gallant. Sag94 hans sagan, spacefilling curves, universitext series, springerverlag, 1994. Proximity problems for which spacefilling curves have been used frequently are approximate nearest neighbors search and finding closest pairs see 99 and the references therein. The way it could have happened and should have happened, but did not happen by hans sagan north carolina state university, department of mathematics, raleigh, nc 276958205, u.
Isbn 9780387942650 this book at amazon print version. Space filling curve, pdf recurrence for pandimensional space filling functions arxiv. A better way of flipping a section is to reverse all of the angles. Hilberts geometric generating process for the construction of a surface filling curve is generalized to lead to a space filling curve and an analyticarithmetic representation of that curve is found in terms of the eight similarity transformations that generate it. Wirth 9, by seeing the recursive nature of the curves, has given a recursive procedure to draw these curves. Representation of a space filling curve by an analytic function 6. Since a two dimensional curve is defined as a continuous mapping from a closed and bounded line segment into. Although most space filling curves are not osgood curves they have positive area but often include infinitely many selfintersections, failing to be jordan curves it is possible to modify the recursive construction of space filling curves or other fractal curves to obtain an osgood curve. Use features like bookmarks, note taking and highlighting while reading spacefilling curves universitext. But worst is that the curve is no longer drawn as a single continuous line. Rapid graph layout using space filling curves chris muelder and kwanliu ma, senior member, ieee abstract network data frequently arises in a wide variety of.
Hans sagan, space filling curves, springerverlag, new york 1994. Computer generation of spacefilling curves several algorithms have been presented for construct ing a hilbert curve. We define exact selfsimilarity of space filling curves on the plane. Use features like bookmarks, note taking and highlighting while reading space filling curves universitext.
Please use this display as a guideline and modify as needed. On space filling curves and the hahnmazurkiewicz theorem alexander kupers abstract. Sagan, hans 1994, spacefilling curves, springerverlag, doi. An elementary proof that schoenbergs spacefilling curve. Heres an excellent curve i first saw in the book brain filling curves the trick is that part of the curve is drawn flipped. An elementary proof that schoenbergs spacefilling curve is. Generation of space filling curves by stochastically independent functions 6. Librarything is a cataloging and social networking site for booklovers.
Introduction 7 a common application of space filling curves is storage and retrieval of multidimensional data in a database 88. Flipped means that instead of just drawing a segment, you instead. The nowhere differentiability of the schoenberg curve 45 46 49 49 51 56 58 60 62 67 69 69 71 74. An interesting book related to space filling curves and. Could someone provide a reference or a sketch of a proof that no differentiable spacefilling curve exists. We use cookies to make interactions with our website easy and meaningful, to better. In this paper we seek to enhance the design of the scaffold architecture by using space filling curves within its interior space. Unfortunately, no comprehensive treatment has ever been attempted other than the gallant effort by w. Flipping by replacing left turns with right turns etc. Spacefilling curves by hans sagan 1994, paperback at the best online prices at ebay. These citations may not conform precisely to your selected citation style.
Download download space filling curves hans sagan pdf read online read online space filling curves hans sagan pdf hilbert space space filling curve algorithm peano curve construction 3d space filling curve hilbert curve function download fulltext pdf. However, there have been no comprehensive treatments of the subject since siepinskys in 1912. For that purpose, we adapt the general definition of exact selfsimilarity on sets, a typical property of fractals, to the. Null 8 was the first to give a nonrecursive algorithm to generate hilbert and other curves. However, formatting rules can vary widely between applications and fields of interest or study. Shchepin the known serpinskyknopp curve 1 maps the segment to an isosceles rectangular triangle in such a way that one half of the segment is mapped to half of the triangle, and the other half the. Hans sagan s 8 research works with 32 citations and 127 reads, including. These are notes on spacefilling curves, looking at a few examples and proving. Here is an ascii textgraphic of the cube filling hilbert curve that was posted to the fracl fractal discussion list. Space filling curves can be employed to accomplish this mapping.
A spacefilling curve is a continuous mapping from a lowerdimensional space into a higherdimensional one. Must every continuous space filling curve be nowhere. Space filling curves universitext by hans sagan and a great selection of related books, art and collectibles available now at. The algorithm, which draws the spacefilling curve, in short is the following for more details have a look at the source code of spacefilling. Stanford libraries official online search tool for books, media, journals, databases, government documents and more. Spacefilling curves universitext kindle edition by sagan, hans. In this section, we give the surprising result that there is a continuous function from the interval 0,1 onto the unit square 0,1.
He studied at the universities of jassy, gottingen, and berlin, and received his doctors degree from the university of jassy in 1926. Space filling curves universitext kindle edition by sagan, hans. Mathematical description in this text we will only deal with two dimensional sfcs. A summary of the ideas, minus technical details but with pointers to technical literature, may be found in my class notes a routing system based on spacefilling curves pdf format, 22 pages.
Proof that no differentiable spacefilling curve exists. A more recent book may be of interest to followers of this thread. In mathematical analysis, a spacefilling curve is a curve whose range contains the entire 2dimensional unit square or more generally an ndimensional unit hypercube. Hans sagan the subject of space filling curves has generated a great deal of interest in the 100 years since the first such curve was discovered by peano. Schoenberg 19031990 was born in galatz, romania, and died in madison, wisconsin. Hans sagan space filling curves with 65 figures springerverlag new york berlin heidelberg london paris tokyo hong kong barcelona budapest. Some reflections on the emergence of space filling curves. Because giuseppe peano 18581932 was the first to discover one, spacefilling curves in the 2dimensional plane are sometimes called peano curves, but that phrase also refers to the peano curve, the specific example of a spacefilling curve found by peano. Pdf an introduction to spacefilling curves kyle byrne. Triple points belong to a relatively simple set, while double points resist easy classification.
These are notes on space lling curves, looking at a few examples and proving the hahnmazurkiewicz theorem. Hans sagan author of boundary and eigenvalue problems in. The subject of space filling curves has fascinated mathematicians for over a century and has intrigued many generations of students of mathematics. Spacefilling curves, puzzling maps between dimensions, have surprising realworld applications.
Space filling curves started their lives as mathematical curiosities, at the end of the nineteenth century. On spacefilling curves and the hahnmazurkiewicz theorem 3 figure 1. A cubefilling hilbert curve university of waterloo. To accompany this is a table of sierpinski indices of the points of a 100 x 100 grid pdf format, 22 pages, with which you can set up your own routing. Hans sagan is the author of boundary and eigenvalue problems in mathematical physics 3. Series universitext subject category mathematical physics and mathematics. Download it once and read it on your kindle device, pc, phones or tablets. Hans sagan sagan, hans used books, rare books and new books. Vector replacement rules allow formulation of the flowsnake planefilling curve as a bijection z z2 and as a surjection 2. Several procedures for drawing particular space filiing curves have been proposed in the literature 2, 7, 10, 12, 15, 16. Space filling curves math 635 5 indexing that would like on top of the onedimensional retrieval system. The concept of space filling curves has been extended to handle polygons. See all books authored by hans sagan, including introduction to the calculus of variations, and boundary and eigenvalue problems in mathematical physics, and more on.
Recurrence for pandimensional spacefilling functions. Space filling curves, recursion, simultaneous grammar, logo, postscript. The performance of space filling curves for dimension reduction. The subject of spacefilling curves has fascinated mathematicians for over a century and has intrigued many generations of students of mathematics. Generation of spacefilling curves by stochastically independent. Could someone provide a reference or a sketch of a proof that no differentiable space filling curve exists. Because giuseppe peano 18581932 was the first to discover one, space filling curves in the 2dimensional plane are sometimes called peano curves, but that phrase also refers to the peano curve, the specific example of a. For the preface and content of the book, visit the following website. The contents throughout is heavily influenced by what is one of the only books to really study the subject, hans sagans space filling curves 1. Working in this area is like skating on the edge of reason.
Besides peanos space filling curve, there are many other famous space filling curves, the book space filling curves wrote by hans sagan introduce several of them. Jordan curves of positive lebesgue measure springer for. The subject of space filling curves has generated a great deal of interest in the 100 years since the first such curve was discovered by peano. This theorem characterizes those subsets of euclidean space that are the image of the unit interval under a continuous space. Cantor, hilbert, moore, knopp, lebesgue, and polya are among the prominent mathematicians who have contributed to the field. However, currently the interior macro pore design of scaffolds have been limited to simple regular shapes of either squares or circles due to limited cad capability. Niedermeier, on multidimensional curves with hilbert property, theory comput.
Hans sagan spacefilling curves with 65 figures springerverlag. The best reference for spacefilling curves is sagans book. What is the state of the art about spacefilling curves in. Also, it introduces some jordan curve with positive lebesgue measure, hope it will help all of us. Hans sagan the subject of space filling curves has generated a great deal of interest since the first such curve was discovered by peano over a century ago. And the whole idea of space filling curves is that you never lift the pen. An elementary proof that schoenbergs space filling curve is nowhere differentiable hans sagan north carolina state university raleigh, nc 27695 up to the end of the nineteenth century and in six, randomly selected, contemporary calculus books a plane curve was and is defined as the graph of a pair of parametric equations xft tee i. Download letters from the teacher volume 2 teachings of the order of christian mystics volume 9 pdf by mrs harriette augusta curtiss download moscow mathematical olympiads, 19931999 msri mathematical circles library pdf by roman fedorov, alexei.