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In one corner of a grid, a single block represents 1, the first square. That can be wrapped on two sides by a strip of three blocks (the next odd number) to make a 2 Ă 2 block: 4, the second square. Adding a further five blocks makes a 3 Ă 3 block: 9, the third square. This
812:
This slim collection of varied visual 'proofs' (a term, it can be argued, loosely applied here) is entertaining and enlightening. I personally find such representations engaging and stimulating aids to that 'aha!' moment when symbolic argument seems not to
684:
We always try to keep clearly in mind what we assume and what we prove. By a 'proof' we mean a sequence of statements each of which is either assumed, or follows from the preceding statements by a rule of deduction, which is itself
84:, because it omits the details of the logical argument it illustrates. However, it can provide valuable intuitions to the viewer that can help them formulate or better understand a true proof.
143:
Rearrangement proof of the
Pythagorean theorem. The uncovered area of gray space remains constant before and after the rearrangement of the triangles: on the left it is shown to equal
214:
476:. A proof without words might imply such an argument, but it does not make one directly, so it cannot take the place of a formal proof where one is required. Rather,
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proofs due to their self-evident nature. When the diagram demonstrates a particular case of a general statement, to be a proof, it must be generalisable.
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run a regular feature titled "Proof without words" containing, as the title suggests, proofs without words. The Art of
Problem Solving and
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However, since most proofs without words are visual in nature, they often provide a reminder or hint of what's missing.
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Benson, Steve; Addington, Susan; Arshavsky, Nina; Cuoco; Al; Goldenberg, E. Paul; Karnowski, Eric (October 6, 2004).
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Benson, Steve; Addington, Susan; Arshavsky, Nina; Cuoco; Al; Goldenberg, E. Paul; Karnowski, Eric (October 6, 2004).
1600:
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by a diagram without any accompanying explanatory text. Such proofs can be considered more elegant than formal or
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use proofs without words as illustrations and teaching aids for ideas that have already been proven formally.
332:. The four triangles can be rearranged within the larger square to split its unused space into two squares of
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in its corners, such that the space in the middle is a diagonal square with an area of
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show how the statement it aims to prove follows totally and inevitably from a set of
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682:. Reading, Massachusetts: Addison-Wesley Publishing Company. p. 94.
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611:, vol. 43, no. 8, American Mathematical Society, p. 527,
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Basing the argument on a geometric picture is not a proof, however...
740:(4th ed.). Houston, Texas: Publish or Perish, Inc. p. 138.
548:
429:) increasingly "stretches" the distribution for increasing values of
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For a proof to be accepted by the mathematical community, it must
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One method of doing so is to visualise a larger square of sides
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proofs, strictly speaking, since details are typically lacking.
453:
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74:
1763:
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954:
796:"Proofs without Words: Exercises in Visual Thinking (review)"
763:
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473:
538:
876:
Proofs without Words II: More
Exercises in Visual Thinking
405:
can also be proven graphically. A dashed curve along the
101:
A proof without words for the sum of odd numbers theorem
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or mathematical statement which can be demonstrated as
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311:
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271:
251:
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121:—can be demonstrated by a proof without words.
859:, Mathematical Association of America, p. 160,
857:Proofs without Words: Exercises in Visual Thinking
834:
770:(Illustrated ed.). Corwin Press. p. 78.
703:(Illustrated ed.). Corwin Press. p. 78.
594:The Moment of Proof : Mathematical Epiphanies
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237:
208:
897:Gulley, Ned (March 4, 2010), Shure, Loren (ed.),
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879:, Mathematical Association of America, pp.
135:Pythagorean theorem § Rearrangement proofs
607:McShane, E. J. (1937), "Jensen's Inequality",
597:, pp. 172â173 (Oxford University Press, 1999).
925:
609:Bulletin of the American Mathematical Society
117:—more specifically, the perfect square
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245:, with four right-angled triangles of sides
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105:The statement that the sum of all positive
80:A proof without words is not the same as a
1117:
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492: â Equality of areas of a sliced disk
417:axis is the corresponding distribution of
802:. The Mathematical Association of America
616:
409:axis is the hypothetical distribution of
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520:
398:A graphical proof of Jensen's inequality
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18:
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125:process can be continued indefinitely.
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16:Mathematical proof expressed visually
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503: â Branch of mathematical logic
460:illustrating proofs without words.
13:
509: â Visual mathematical proofs
14:
2699:
794:Schulte, Tom (January 12, 2011).
413:, while a dashed curve along the
209:{\displaystyle a^{2}+b^{2}=c^{2}}
2651:
658:, USA Mathematical Talent Search
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618:10.1090/S0002-9904-1937-06588-8
449:The College Mathematics Journal
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1:
2612:History of mathematical logic
822:
715:Proofs without words are not
216:can be proven without words.
2537:Primitive recursive function
31:) that the sum of the first
7:
483:
421:values. The convex mapping
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65:) is an illustration of an
23:Proof without words of the
10:
2704:
1601:SchröderâBernstein theorem
1328:Monadic predicate calculus
987:Foundations of mathematics
132:
113: − 1 is a
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2634:Philosophy of mathematics
2583:Automated theorem proving
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1708:Von NeumannâBernaysâGödel
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873:Nelsen, Roger B. (2000),
855:Nelsen, Roger B. (1997),
837:The Mathematical Universe
496:Philosophy of mathematics
464:Compared to formal proofs
636:, Art of Problem Solving
513:
436:
2284:Self-verifying theories
2105:Tarski's axiomatization
1056:Tarski's undefinability
1051:incompleteness theorems
841:, John Wiley and Sons,
75:mathematically rigorous
2658:Mathematics portal
2269:Proof of impossibility
1917:propositional variable
1227:Propositional calculus
555:Retrieved on 2008-6-20
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2527:Kolmogorov complexity
2480:Computably enumerable
2380:Model complete theory
2172:Principia Mathematica
1232:Propositional formula
1061:BanachâTarski paradox
544:"Proof without Words"
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379:{\displaystyle b^{2}}
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352:{\displaystyle a^{2}}
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325:{\displaystyle c^{2}}
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1189:Square of opposition
1047:Gödel's completeness
900:Nicomachus's Theorem
443:Mathematics Magazine
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2683:Mathematical proofs
2678:Proof without words
2629:Mathematical object
2520:P versus NP problem
2485:Computable function
2279:Reverse mathematics
2205:Logical consequence
2082:primitive recursive
2077:elementary function
1850:Free/bound variable
1703:TarskiâGrothendieck
1222:Logical connectives
1152:Logical equivalence
1002:Logical consequence
403:Jensen's inequality
390:Jensen's inequality
238:{\displaystyle a+b}
161:Pythagorean theorem
149:, and on the right
129:Pythagorean theorem
59:proof without words
2427:Transfer principle
2390:Semantics of logic
2375:Categorical theory
2351:Non-standard model
1865:Logical connective
992:Information theory
941:Mathematical logic
541:Weisstein, Eric W.
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93:Sum of odd numbers
82:mathematical proof
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25:Nicomachus theorem
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2597:Abstract category
2400:Theories of truth
2210:Rule of inference
2200:Natural deduction
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1431:Cartesian product
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1242:Many-valued logic
1217:Boolean functions
1100:Russell's paradox
1075:diagonal argument
972:First-order logic
866:978-0-88385-700-7
747:978-0-914098-91-1
680:Basic Mathematics
655:Gallery of Proofs
633:Gallery of Proofs
298:{\displaystyle c}
278:{\displaystyle b}
258:{\displaystyle a}
48:triangular number
2695:
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2607:History of logic
2602:Category of sets
2495:Decision problem
2274:Ordinal analysis
2215:Sequent calculus
2113:Boolean algebras
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1998:logical/constant
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1661:ZermeloâFraenkel
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2578:Algebraic logic
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2532:Lambda calculus
2470:Church encoding
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2432:Truth predicate
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1666:axiom of choice
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1611:Gödel numbering
1551: and
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1212:Boolean algebra
1198:
1162:Equiconsistency
1127:Classical logic
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1085:Halting problem
1073: and
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1037: and
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1031:Theorems (
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831:Dunham, William
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2385:Satisfiability
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2123:minimal axioms
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1386:Extensionality
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1381:Ordinal number
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512:
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504:
498:
493:
485:
482:
478:mathematicians
465:
462:
438:
435:
391:
388:
373:
369:
346:
342:
319:
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274:
254:
234:
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228:
203:
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177:
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133:Main article:
130:
127:
115:perfect square
94:
91:
89:
86:
15:
9:
6:
4:
3:
2:
2700:
2689:
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2576:
2574:
2571:
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2564:
2558:
2555:
2553:
2550:
2548:
2547:Recursive set
2545:
2543:
2540:
2538:
2535:
2533:
2530:
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2525:
2521:
2518:
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2366:
2365:
2364:
2361:
2357:
2356:of arithmetic
2354:
2353:
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2337:
2335:
2332:
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2327:
2325:
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2297:
2295:
2291:
2285:
2282:
2280:
2277:
2275:
2272:
2270:
2267:
2264:
2263:from ZFC
2260:
2257:
2255:
2252:
2246:
2243:
2242:
2241:
2238:
2236:
2233:
2231:
2228:
2227:
2226:
2223:
2221:
2218:
2216:
2213:
2211:
2208:
2206:
2203:
2201:
2198:
2196:
2193:
2192:
2190:
2188:
2184:
2174:
2173:
2169:
2168:
2163:
2162:non-Euclidean
2160:
2156:
2153:
2151:
2148:
2146:
2145:
2141:
2140:
2138:
2135:
2134:
2132:
2128:
2124:
2121:
2119:
2116:
2115:
2114:
2110:
2106:
2103:
2102:
2101:
2097:
2093:
2090:
2088:
2085:
2083:
2080:
2078:
2075:
2073:
2070:
2068:
2065:
2064:
2062:
2058:
2057:
2055:
2050:
2044:
2039:Example
2036:
2028:
2023:
2022:
2021:
2018:
2016:
2013:
2009:
2006:
2004:
2001:
1999:
1996:
1994:
1991:
1990:
1989:
1986:
1984:
1981:
1979:
1976:
1974:
1971:
1967:
1964:
1962:
1959:
1958:
1957:
1954:
1950:
1947:
1945:
1942:
1940:
1937:
1935:
1932:
1931:
1930:
1927:
1925:
1922:
1918:
1915:
1913:
1910:
1908:
1905:
1904:
1903:
1900:
1896:
1893:
1891:
1888:
1886:
1883:
1881:
1878:
1876:
1873:
1871:
1868:
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1826:
1825:
1824:
1821:
1819:
1816:
1814:
1811:
1809:
1806:
1802:
1799:
1797:
1796:by definition
1794:
1793:
1792:
1789:
1785:
1782:
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1780:
1777:
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1709:
1706:
1704:
1701:
1699:
1696:
1694:
1691:
1689:
1686:
1684:
1683:KripkeâPlatek
1681:
1679:
1676:
1672:
1669:
1667:
1664:
1663:
1662:
1659:
1658:
1656:
1652:
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1639:
1636:
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1614:
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1602:
1599:
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1588:
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1568:
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1566:
1562:
1559:
1558:
1556:
1554:
1550:
1546:
1538:
1535:
1533:
1530:
1528:
1527:constructible
1525:
1524:
1523:
1520:
1518:
1515:
1513:
1510:
1508:
1505:
1503:
1500:
1498:
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1468:
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1442:
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1434:
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1417:
1414:
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1411:
1407:
1404:
1402:
1399:
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1382:
1379:
1377:
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1369:
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1321:
1319:
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1314:
1311:
1309:
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1301:
1297:
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1235:
1233:
1230:
1228:
1225:
1223:
1220:
1218:
1215:
1213:
1210:
1209:
1207:
1205:
1204:Propositional
1201:
1195:
1192:
1190:
1187:
1185:
1182:
1180:
1177:
1175:
1172:
1170:
1167:
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1160:
1159:
1158:
1155:
1153:
1150:
1148:
1145:
1143:
1140:
1138:
1135:
1133:
1132:Logical truth
1130:
1128:
1125:
1124:
1122:
1120:
1116:
1113:
1111:
1107:
1101:
1098:
1096:
1093:
1091:
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1086:
1083:
1081:
1078:
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1072:
1068:
1064:
1062:
1059:
1057:
1054:
1052:
1048:
1045:
1044:
1042:
1040:
1034:
1029:
1023:
1020:
1018:
1015:
1013:
1010:
1008:
1005:
1003:
1000:
998:
995:
993:
990:
988:
985:
983:
980:
978:
975:
973:
970:
968:
965:
961:
958:
957:
956:
953:
952:
950:
946:
942:
935:
930:
928:
923:
921:
916:
915:
912:
902:
901:
895:
892:
890:0-88385-721-9
886:
882:
878:
877:
871:
868:
862:
858:
853:
850:
848:0-471-53656-3
844:
839:
838:
832:
828:
827:
814:
801:
797:
790:
783:
779:
777:9781412905206
773:
769:
768:
760:
753:
749:
743:
739:
738:
733:
727:
720:
716:
712:
710:9781412905206
706:
702:
701:
693:
686:
681:
677:
671:
657:
656:
649:
635:
634:
627:
619:
614:
610:
603:
596:
595:
588:
581:
576:
570:, p. 121
569:
564:
562:
551:
550:
545:
542:
535:
529:, p. 120
528:
523:
519:
508:
505:
502:
499:
497:
494:
491:
490:Pizza theorem
488:
487:
481:
479:
475:
471:
461:
459:
456:websites run
455:
451:
450:
445:
444:
434:
432:
428:
424:
420:
416:
412:
408:
404:
396:
387:
371:
367:
344:
340:
317:
313:
292:
272:
252:
232:
229:
226:
217:
201:
197:
193:
188:
184:
180:
175:
171:
162:
154:
153:
148:
147:
141:
136:
126:
122:
120:
116:
112:
108:
99:
85:
83:
78:
76:
72:
68:
64:
60:
56:
49:
41:
37:
30:
29:Gulley (2010)
26:
21:
2649:
2447:Ultraproduct
2294:Model theory
2259:Independence
2195:Formal proof
2187:Proof theory
2170:
2143:
2100:real numbers
2072:second-order
1983:Substitution
1860:Metalanguage
1801:conservative
1774:Axiom schema
1718:Constructive
1688:MorseâKelley
1654:Set theories
1633:Aleph number
1626:inaccessible
1532:Grothendieck
1416:intersection
1303:Higher-order
1291:Second-order
1237:Truth tables
1194:Venn diagram
977:Formal proof
899:
875:
856:
836:
811:
804:. Retrieved
799:
789:
781:
766:
759:
751:
736:
726:
714:
699:
692:
683:
679:
670:
660:, retrieved
654:
648:
638:, retrieved
632:
626:
608:
602:
592:
587:
575:
547:
534:
522:
501:Proof theory
467:
458:Java applets
447:
441:
440:
430:
426:
422:
418:
414:
410:
406:
401:
218:
158:
151:
150:
145:
144:
123:
118:
110:
104:
79:
71:self-evident
63:visual proof
62:
58:
52:
2557:Type theory
2505:undecidable
2437:Truth value
2324:equivalence
2003:non-logical
1616:Enumeration
1606:Isomorphism
1553:cardinality
1537:Von Neumann
1502:Ultrafilter
1467:Uncountable
1401:equivalence
1318:Quantifiers
1308:Fixed-point
1277:First-order
1157:Consistency
1142:Proposition
1119:Traditional
1090:Lindström's
1080:Compactness
1022:Type theory
967:Cardinality
806:October 26,
800:MAA Reviews
676:Lang, Serge
582:, p. 3
580:Nelsen 1997
568:Dunham 1994
527:Dunham 1994
474:assumptions
107:odd numbers
55:mathematics
2672:Categories
2368:elementary
2061:arithmetic
1929:Quantifier
1907:functional
1779:Expression
1497:Transitive
1441:identities
1426:complement
1359:hereditary
1342:Set theory
823:References
662:2015-05-28
640:2015-05-28
2639:Supertask
2542:Recursion
2500:decidable
2334:saturated
2312:of models
2235:deductive
2230:axiomatic
2150:Hilbert's
2137:Euclidean
2118:canonical
2041:axiomatic
1973:Signature
1902:Predicate
1791:Extension
1713:Ackermann
1638:Operation
1517:Universal
1507:Recursive
1482:Singleton
1477:Inhabited
1462:Countable
1452:Types of
1436:power set
1406:partition
1323:Predicate
1269:Predicate
1184:Syllogism
1174:Soundness
1147:Inference
1137:Tautology
1039:paradoxes
549:MathWorld
470:logically
2624:Logicism
2617:timeline
2593:Concrete
2452:Validity
2422:T-schema
2415:Kripke's
2410:Tarski's
2405:semantic
2395:Strength
2344:submodel
2339:spectrum
2307:function
2155:Tarski's
2144:Elements
2131:geometry
2087:Robinson
2008:variable
1993:function
1966:spectrum
1956:Sentence
1912:variable
1855:Language
1808:Relation
1769:Automata
1759:Alphabet
1743:language
1597:-jection
1575:codomain
1561:Function
1522:Universe
1492:Infinite
1396:Relation
1179:Validity
1169:Argument
1067:theorem,
833:(1994),
813:clarify.
737:Calculus
734:(2008).
685:assumed.
678:(1971).
484:See also
88:Examples
67:identity
2566:Related
2363:Diagram
2261: (
2240:Hilbert
2225:Systems
2220:Theorem
2098:of the
2043:systems
1823:Formula
1818:Grammar
1734: (
1678:General
1391:Forcing
1376:Element
1296:Monadic
1071:paradox
1012:Theorem
948:General
109:up to 2
42:of the
38:is the
2329:finite
2092:Skolem
2045:
2020:Theory
1988:Symbol
1978:String
1961:atomic
1838:ground
1833:closed
1828:atomic
1784:ground
1747:syntax
1643:binary
1570:domain
1487:Finite
1252:finite
1110:Logics
1069:
1017:Theory
887:
863:
845:
774:
744:
717:really
707:
454:USAMTS
40:square
2319:Model
2067:Peano
1924:Proof
1764:Arity
1693:Naive
1580:image
1512:Fuzzy
1472:Empty
1421:union
1366:Class
1007:Model
997:Lemma
955:Axiom
514:Notes
437:Usage
163:that
152:aÂČ+bÂČ
36:cubes
2442:Type
2245:list
2049:list
2026:list
2015:Term
1949:rank
1843:open
1737:list
1549:Maps
1454:sets
1313:Free
1283:list
1033:list
960:list
885:ISBN
861:ISBN
843:ISBN
808:2022
772:ISBN
742:ISBN
705:ISBN
446:and
359:and
285:and
159:The
61:(or
57:, a
2129:of
2111:of
2059:of
1591:Sur
1565:Map
1372:Ur-
1354:Set
881:142
613:doi
53:In
46:th
2674::
2515:NP
2139::
2133::
2063::
1740:),
1595:Bi
1587:In
883:,
810:.
798:.
780:.
750:.
713:.
560:^
546:.
433:.
386:.
265:,
146:cÂČ
2595:/
2510:P
2265:)
2051:)
2047:(
1944:â
1939:!
1934:â
1895:=
1890:â
1885:â
1880:â§
1875:âš
1870:ÂŹ
1593:/
1589:/
1563:/
1374:)
1370:(
1257:â
1247:3
1035:)
933:e
926:t
919:v
905:.
615::
552:.
431:X
427:X
425:(
423:Y
419:Y
415:Y
411:X
407:X
372:2
368:b
345:2
341:a
318:2
314:c
293:c
273:b
253:a
233:b
230:+
227:a
202:2
198:c
194:=
189:2
185:b
181:+
176:2
172:a
155:.
119:n
111:n
44:n
33:n
27:(
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