[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-seo-567644-105":3,"detail-sidebar-cat-0-en-105":80,"doc-detail-567644-en":130},{"code":4,"msg":5,"data":6},0,"ok",{"site_id":7,"language":8,"slug":9,"title":10,"keywords":11,"description":12,"schema_data":13,"social_meta":73,"head_meta":75,"extra_data":77,"updated_unix":79},105,"en","topics-in-mathematics-equivalent-and-equidecomposable-figures","Topics in Mathematics - Equivalent and Equidecomposable Figures","","Introduces fundamental problems in the theories of area and volume through the lens of geometric dissection. It studies whether two polygons with equal areas can be partitioned into finitely many pieces and rearranged to form each other, then extends the analogous question to solid figures. The booklet explains key theorems and methods, while noting the mathematical prerequisites and varying proof difficulty across its chapters.",{"@graph":14,"@context":72},[15,34,55],{"@type":16,"itemListElement":17},"BreadcrumbList",[18,23,27,31],{"item":19,"name":20,"@type":21,"position":22},"https://docshare.wps.com","Home","ListItem",1,{"item":24,"name":25,"@type":21,"position":26},"https://docshare.wps.com/document/","Document",2,{"item":28,"name":29,"@type":21,"position":30},"https://docshare.wps.com/document/research-report/","Research & Report",3,{"item":32,"name":10,"@type":21,"position":33},"https://docshare.wps.com/document/topics-in-mathematics-equivalent-and-equidecomposable-figures/567644/",4,{"url":32,"name":10,"@type":35,"image":36,"author":41,"headline":10,"publisher":44,"fileFormat":47,"inLanguage":8,"description":12,"dateModified":48,"datePublished":49,"encodingFormat":47,"isAccessibleForFree":50,"interactionStatistic":51},"DigitalDocument",{"url":37,"@type":38,"width":39,"height":40},"https://docshare.wps.com/thumbnails/topics-in-mathematics-equivalent-and-equidecomposable-figures/567644.png","ImageObject",300,407,{"name":42,"@type":43},"Connor ","Person",{"url":19,"name":45,"@type":46},"DocShare","Organization","application/pdf","2026-10-16","2026-10-13",true,{"@type":52,"interactionType":53,"userInteractionCount":26},"InteractionCounter",{"@type":54},"ViewAction",{"@type":56,"mainEntity":57},"FAQPage",[58,64,68],{"name":59,"@type":60,"acceptedAnswer":61},"What central question does the booklet address for plane figures?","Question",{"text":62,"@type":63},"It asks whether two polygons with equal areas can be dissected into a finite number of parts and rearranged to form the other.","Answer",{"name":65,"@type":60,"acceptedAnswer":66},"How does the booklet extend the plane-figure problem to solids?",{"text":67,"@type":63},"It considers the analogous equidecomposability problem for solid figures (polyhedra), focusing on how volumes and dissections relate.",{"name":69,"@type":60,"acceptedAnswer":70},"What mathematical background does the reader need for the different chapters?",{"text":71,"@type":63},"The first four chapters require about one year of algebra and a half-year of plane geometry, while the last two chapters also need solid geometry and trigonometry.","https://schema.org",{"og:url":32,"og:type":74,"og:title":10,"og:site_name":45,"og:description":12},"article",{"robots":76,"canonical":32},"index,follow",{"doc_id":78,"site_id":7},567644,1792190414,{"code":4,"msg":81,"data":82},"success",[83,87,91,95,100,105,110,114,119,122,126],{"id":22,"doc_module":4,"doc_module_name":25,"category_name":84,"show_sort_weight":85,"slug":86},"Story & Novel",90,"story-novel",{"id":26,"doc_module":4,"doc_module_name":25,"category_name":88,"show_sort_weight":89,"slug":90},"Literature",80,"literature",{"id":33,"doc_module":4,"doc_module_name":25,"category_name":92,"show_sort_weight":93,"slug":94},"Exam",70,"exam",{"id":96,"doc_module":4,"doc_module_name":25,"category_name":97,"show_sort_weight":98,"slug":99},5,"Comic",60,"comic",{"id":101,"doc_module":4,"doc_module_name":25,"category_name":102,"show_sort_weight":103,"slug":104},6,"Technology",50,"technology",{"id":106,"doc_module":4,"doc_module_name":25,"category_name":107,"show_sort_weight":108,"slug":109},7,"Healthcare",40,"healthcare",{"id":111,"doc_module":4,"doc_module_name":25,"category_name":29,"show_sort_weight":112,"slug":113},8,30,"research-report",{"id":115,"doc_module":4,"doc_module_name":25,"category_name":116,"show_sort_weight":117,"slug":118},9,"Religion & Spirituality",20,"religion-spirituality",{"id":117,"doc_module":4,"doc_module_name":25,"category_name":120,"show_sort_weight":117,"slug":121},"World Cup","world-cup",{"id":123,"doc_module":4,"doc_module_name":25,"category_name":124,"show_sort_weight":123,"slug":125},10,"Lifestyle","lifestyle",{"id":127,"doc_module":4,"doc_module_name":25,"category_name":128,"show_sort_weight":96,"slug":129},19,"General","general",{"code":4,"msg":81,"data":131},{"doc_id":78,"user_id":132,"nickname":42,"user_avatar":133,"doc_module":4,"category_id":111,"category_name":29,"doc_title":10,"doc_description":12,"doc_content":134,"file_id":135,"file_url":136,"file_type":137,"file_size":138,"view_count":26,"is_deleted":4,"is_public":22,"is_downloadable":22,"audit_status":22,"page_count":139,"language":140,"language_code":8,"site_id":7,"html_lang":8,"table_of_contents":141,"faqs":142,"seo_title":143,"seo_description":12,"update_tm":144,"read_time":145},687207022233,"https://ap-avatar.wpscdn.com/davatar_155a257f0dc6eb9ab79c44ca47cae57d","# TOPICS IN MATHEMATICS\n\nTRANSLATED FROM THE RUSSIANEQUIVALENT ANDEQUIDECOMPOSABLEFIGURESV.G.BOLTYANSKII  \n## D.C.HEATH AND COMPANY\n\nToPICS IN MATHEMATICs is a series of booklets translated andadapted from the Russian series Popular Lectures in Mathe-matics.The purpose of these booklets is to introduce thereader to various aspects of mathematical thought and toengage him in mathematical activity of a kind that fostershabits leading to independent creative work.The series willmake available to students of mathematics at various levels,as well as to other interested readers,valuable supplementarymaterial to further their mathematical knowledge anddevelopment.  \nSome of the booklets contain expositions of concepts andmethods in greater depth than is usually found in standardtexts.Others are introductions from an elementary point ofview to significant chapters of modern or higher mathematicsEach booklet is largely self-contained.  \nThe series Popular Lectures in Mathematics occupies acentral position in the extensive extracurricular literature inmathematics in the Soviet Union.It was started in 1950,andby 1961 more than thirty volumes had appeared.Individualbooklets are,for the most part,elaborations by prominentmathematicians of special lectures given by them at MoscowState University and other universities.The booklets arewidely used in mathematics clubs in the Soviet Union.  \nThese American editions have been prepared by the Sur-vey of Recent East European Mathematical Literature atthe University of Chicago under a grant from the NationalScience Foundation.  \nTrTLES OF the series Topics in Mathematics are listed on theinside back cover.  \nEquivalent and  \nEquidecomposable FiguresV.G.Boltyanskii  \nTranslated and adapted from the first Russian edition(1956)byALFRED K.HENN and CHARLES E.WATTS  \n## SURVEY OFRECENT EAST EUROPEAN MATHEMATICAL LITERATURE\n\nA project conducted byALFRED L.PUTNAM and IZAAK WIRSZUP  \nDepartment of Mathematics,The University of Chicago,under agrant from the National Science Foundation  \nLibrary of Congress Catalog Card Number:61-17839  \nCopyright ◎1963 by THE UNIVERSITY OF CHICAGO  \nNo part of the material covered by this copyright may be repro-duced in any form without written permission of the publisher.Printed in the United States of America  \nPREFACE TO THE AMERICAN EDITION  \nTHIS BOOKLET deals with a class of problems fundamental to thetheories of area and volume.In the case of plane figures the centralproblem considered stems from the following question:If twopolygons have equal areas,is it possible to dissect one of theminto a finite number of parts which can be rearranged to form theother?The second part of the booklet is concerned with the anal-ogous problem for solid figures.In view of the elementary natureof the topics themselves,it may seem surprising that some of thetheorems proved in this booklet are the product of comparativelyrecent research.  \nFor the first four chapters,the reader needs only the backgroundof one year of algebra and a half-year of plane geometry.Solidgeometry and trigonometry are needed for the last two chapters.Some of the proofs toward the end of the booklet are rather difficult,and on first reading the student may wish to learn only the state-ment of these theorems.  \nCONTENTS  \nCHAPTER 1.The Bolyai-Gerwin Theorem;Equidecomposability of Polygons  \n1  \n1.Decomposition method  \nI3  \n2.The Bolyai-Gerwin theorem  \n3.Complementation method  \n8  \nCHAPTER 2.The Hadwiger-Glur Theorem  \n11  \n4.Motions11  \n5.The Hadwiger-Glur theorem16  \nCHAPTER 3.Equidecomposability and the Concept ofAdditive Invariants  \n21  \n6.The additive invariant Ji(M)21  \n7.T-equidecomposability  \n22  \n8.Properties of the invariant J₂(M)23  \n9.Centrally symmetric polygons25  \nCHAPTER 4.Equidecomposability and the Concept ofGroups  \n28  \n10.Groups  \n28  \n11.Groups of motions  \n30  \n12.A property of the group S(optional)  \n31  \nCHAPTER 5.The Theorems of Dehn and Hadwiger forPolyhedra  \n37  \n13.Equidecomposabl","cbCaifzLSWmqnNQ8","https://ap.wps.com/l/cbCaifzLSWmqnNQ8","pdf",3481934,84,"English","# Preface to the American Edition\n# Contents\n## Chapter 1: The Bolyai-Gerwin Theorem; Equidecomposability of Polygons\n## Chapter 2: The Hadwiger-Glur Theorem\n## Chapter 3: Equidecomposability and the Concept of Additive Invariants\n## Chapter 4: Equidecomposability and the Concept of Groups\n## Chapter 5: The Theorems of Dehn and Hadwiger for Polyhedra\n## Chapter 6: Methods for Calculating Volumes\n## Appendix\n## Bibliography","[{\"question\":\"What central question does the booklet address for plane figures?\",\"answer\":\"It asks whether two polygons with equal areas can be dissected into a finite number of parts and rearranged to form the other.\"},{\"question\":\"How does the booklet extend the plane-figure problem to solids?\",\"answer\":\"It considers the analogous equidecomposability problem for solid figures (polyhedra), focusing on how volumes and dissections relate.\"},{\"question\":\"What mathematical background does the reader need for the different chapters?\",\"answer\":\"The first four chapters require about one year of algebra and a half-year of plane geometry, while the last two chapters also need solid geometry and trigonometry.\"}]","Topics in Mathematics - Equivalent and Equidecomposable Figures | PDF",1791867303,212]