D # = JO ... C C Part 1 JT B SFMBUJPO #
. JOWPMWFT ëSTU = O UP B TFU " " O + GOLDEN UIFSF JT BO FMFNFOU ! " GSPN B TFU JO BSF JO UIF SFMBUJPO UIFO →∞ G B D O
JT JO UIF SFMBUJPO B C MJN to
1MFBTF TFF FOEOPUFT GPS TPNF DPNNFOUT B = " QBSBMMFMPHSBN JT B RVBESJMBUFSBM XJUI IBMG F UIBU TBUJTëFT UIF GPMMPXJOH DPOEJUJPOT BOE BQQFBST UP CF B QFSTJTUFOU CFMJFG JO NBUIFNBUJDBM # C
" GVODUJPO UP B ning; golden rectangle; golden rhombus; rhombus; golden rectangle; golden ning; f de constructive UFYUCPPLT BOE NBUIFNBUJDT UFBDIJOH UIBU HPPE QSBDUJDF NPTUMZ TFF GPPUOPUF IFSF " GPS FBDI FMFNFOU TVDI UIBU JG 1BSBMMFMPHSBN UVSO TZNNFUSZ PO UIJT EFëOJUJPO ɨF OVNCFS 'VODUJPO GSPN *U JT OPU CFJOH DMBJNFE IFSF UIBU BMM UFYUCPPLT BOE UFBDIJOH QSBDUJDFT GPMMPX UIF Keywords: golden parallelogram • CFGPSF FYBNQMFT PG UIF DPODFQUFYQMPSFE BOE NPTUMZ JUT EFEVDUJWFMZ QSPQFSUJFT CVU BSF TPNFUJNFT GVSUIFS FYQFSJNFOUBMMZ BT XFMM 5ZQJDBMMZ B EFëOJUJPO JT ëSTU QSPWJEFE BT GPMMPXT QSPWJEJOH TUVEFOUT XJUI B DPODJTF EFëOJUJPO PG B DPODFQU BQQSPBDI PVUMJOFE IFSF BT UIFSF BSFUIBU TPNF TDIPPM TFSJPVTMZ UFYUCPPLT BUUFNQU TVDI BT UP 4FSSBUSJBOHMFT BDUJWFMZ BOE JOWPMWF TUVEFOUT RVBESJMBUFSBMT JO UIFNTFMWFTDPVSTFT EFëOJOH OPXBEBZT BOE GPS "MTP DMBTTJGZJOH FYBNQMF JO TPNFVTFE CFGPSF HSBQIJDBM NPTU JOUSPEVDJOH BOE B JOUSPEVDUPSZ OVNFSJDBM GPSNBM BQQSPBDIFT MJNJU DBMDVMVT UP EFëOJUJPO BSF UIF PG DVSWF EJêFSFOUJBUJPO PG BT B B GVODUJPOB UBOHFOU PS QBSUJDVMBS GPS QPJOU EFUFSNJOJOH JUT JOTUBOUBOFPVT SBUF PG DIBOHF BU • • 5 GOLDEN
Vol. 6, No. 1, March 2017
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QUADRILATERALS RECTANGLE
and Beyond MICHAEL DE VILLIERS An example of constructive defining: of constructive example An a From At Right Angles 64
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PG NBUIFNBUJDT BT BO ABCTPMVUJTU TDJFODF &SOFTU ◦ Y − = φ, ∴ Y = ◦ .◦. Y + φ ≈ *O HFOFSBM UIFSF BSF FTTFOUJBMMZ UXP EJêFSFOU XBZT "O BDDVSBUF DPOTUSVDUJPO PG B AHPMEFO SIPNCVT PG EFëOJOH NBUIFNBUJDBM DPODFQUT OBNFMZ GVMëMMJOH UIJT BOHMF DPOEJUJPO JT TIPXO JO 'JHVSF EFTDSJQUJWF B QPTUFSJPSJ BOE DPOTUSVDUJWF B QSJPSJ EFëOJOH %FTDSJQUJWF EFëOJUJPOT TZTUFNBUJ[F BMSFBEZ FYJTUJOH LOPXMFEHF XIFSFBT DPOTUSVDUJWF EFëOJUJPOT QSPEVDF OFX LOPXMFEHF 'SFVEFOUIBM ɨF QVSQPTF PG UIJT BSUJDMF JT UP IFVSJTUJDBMMZ JMMVTUSBUF UIF QSPDFTT PG DPOTUSVDUJWF EFëOJOH JO SFMBUJPO UP B SFDFOU FYQMPSBUJPO CZ NZTFMG PG UIF DPODFQU PG B AHPMEFO SFDUBOHMF BOE JUT FYUFOTJPO UP 'JHVSF (PMEFO SIPNCVT XJUI BOHMFT JO SBUJP QIJ B AHPMEFO SIPNCVT AHPMEFO QBSBMMFMPHSBN AHPMEFO USBQF[JVN AHPMEFO LJUF FUD ɨPVHI ɨPVHI UIJT QBSUJDVMBS SIPNCVT MPPLT SFBTPOBCMZ UIFTF FYBNQMFT BSF NBUIFNBUJDBMMZ FMFNFOUBSZ JU WJTVBMMZ BQQFBMJOH * XPOEFSFE IPX FMTF POF NJHIU JT IPQFE UIBU UIFJS EJTDVTTJPO XJMM JMMVNJOBUF UIF SFBTPOBCMZ PCUBJO PS EFëOF UIF DPODFQU PG B EFFQFS QSPDFTT PG DPOTUSVDUJWF EFëOJOH HPMEFO SIPNCVT 4JODF B SFDUBOHMF JT DZDMJD BOE B SIPNCVT IBT BO JOTDSJCFE DJSDMF * IJU VQPO UIF $POTUSVDUJWFMZ %FíOJOH B A(PMEFO 3IPNCVT JEFB PG TUBSUJOH XJUI B HPMEFO SFDUBOHMF &'() &) i <DzF> BMHPSJUINJDBMMZ DPOTUSVDUJWF BOE XJUI &' = φ BOE JUT DJSDVNDJSDMF BOE UIFO DSFBUJWF EFíOJUJPO NPEFMT OFX PCKFDUT DPOTUSVDUJOH UIF SIPNCVT "#$% XJUI TJEFT PVU PG GBNJMJBS POFTw UBOHFOU UP UIF DJSDVNDJSDMF BU UIF WFSUJDFT PG UIF o )BOT 'SFVEFOUIBM SFDUBOHMF /PUF UIBU JU GPMMPXT EJSFDUMZ GSPN UIF TZNNFUSZ PG UIF SFDUBOHMF &'() UIBU "#$% JT B $POTUSVDUJWF B QSJPSJ EFëOJOH UBLFT QMBDF XIFO B SIPNCVT .VDI UP NZ TVSQSJTFE EFMJHIU * OPX HJWFO EFëOJUJPO PG B DPODFQU JT DIBOHFE UISPVHI GPVOE UISPVHI BDDVSBUF DPOTUSVDUJPO BOE UIF FYDMVTJPO HFOFSBMJ[BUJPO TQFDJBMJ[BUJPO NFBTVSFNFOU XJUI EZOBNJD HFPNFUSZ TPGUXBSF BT SFQMBDFNFOU PS BEEJUJPO PG QSPQFSUJFT UP UIF TIPXO JO 'JHVSF UIBU UIPVHI UIF BOHMFT XFSF OP EFëOJUJPO TP UIBU B OFX DPODFQU JT DPOTUSVDUFE JO MPOHFS JO UIF SBUJP QIJ UIF EJBHPOBMT GPS UIJT UIF QSPDFTT SIPNCVT OPX XFSF
At Right Angles | Vol. 6, No. 1, March 2017 65 BFTUIFUJD QPJOU PG WJFX GPPUOPUF XF DPVME UIFSFGPSF DIPPTF FJUIFS POF PG UIF BGPSFNFOUJPOFE QPTTJCJMJUJFT BT PVS EFëOJUJPO )PXFWFS JU TFFNT UIBU B CFUUFS BSHVNFOU DBO CF NBEF GPS UIF TFDPOE EFëOJUJPO PG B AHPMEFO SIPNCVT TJODF JU TIPXT B OJDF EJSFDU DPOOFDUJPO XJUI UIF HPMEFO SFDUBOHMF "MTP OPUF UIBU UIF TFDPOE EFëOJUJPO DBO CF TUBUFE JO FJUIFS PG UIF GPMMPXJOH FRVJWBMFOU GPSNT B RVBESJMBUFSBM XJUI TJEFT DPOTUSVDUFE UBOHFOUJBM UP UIF DJSDVNDJSDMF BOE BU UIF WFSUJDFT PG B HPMEFO SFDUBOHMF BT JMMVTUSBUFE JO 'JHVSF PS NPSF TJNQMZ BT B SIPNCVT XJUI EJBHPOBMT JO UIF SBUJP PG φ GPPUOPUF ɨF DBTF GPS UIF TFDPOE EFëOJUJPO JT GVSUIFS TUSFOHUIFOFE CZ UIF OJDF EVBMJUZ JMMVTUSBUFE CFUXFFO UIF HPMEFO SFDUBOHMF BOE HPMEFO SIPNCVT JO 'JHVSF XIJDI TIPXT UIFJS SFTQFDUJWF NJEQPJOU 'JHVSF (PMEFO SIPNCVT XJUI EJBHPOBMT JO SBUJP QIJ RVBESJMBUFSBMT HFOFSBMMZ DBMMFE A7BSJHOPO *U JT OPU EJïDVMU UP FYQMBJO XIZ QSPWF UIBU UIF QBSBMMFMPHSBNT 4JODF UIF EJBHPOBMT PG UIF HPMEFO EJBHPOBMT PG SIPNCVT "#$% BSF JO UIF SBUJP φ SFDUBOHMF BSF FRVBM JU GPMMPXT UIBU JUT $MFBSMZ USJBOHMFT "#, BOE ,&. BSF TJNJMBS GSPN DPSSFTQPOEJOH 7BSJHOPO QBSBMMFMPHSBN JT B ", ,. SIPNCVT CVU TJODF JUT EJBHPOBMT BSF FRVBM UP UIF XIJDI GPMMPXT UIBU #, = &. #VU ,. = -& TP ", -& TJEFT PG UIF HPMEFO SFDUBOHMF UIFZ BSF BMTP JO UIF #, = &. #VU UIFTF MFOHUIT (",, #,); (-&, &.) BSF SFTQFDUJWFMZ IBMG UIF MFOHUIT PG UIF EJBHPOBMT PG HPMEFO SBUJP BOE UIFSFGPSF UIF SIPNCVT JT B UIF SIPNCVT BOE UIF TJEFT PG UIF SFDUBOHMF IFODF HPMEFO SIPNCVT 4JNJMBSMZ JU GPMMPXT UIBU UIF UIF SFTVMU GPMMPXT GSPN UIF QSPQFSUZ PG UIF HPMEFO 7BSJHOPO QBSBMMFMPHSBN PG UIF HPMEFO SIPNCVT JT B SFDUBOHMF "#$% HPMEFO SFDUBOHMF ɨF TJ[F PG UIF BOHMFT PG UIF HPMEFO SIPNCVT JO $POTUSVDUJWFMZ %FíOJOH B A(PMEFO 'JHVSF DBO FBTJMZ CF EFUFSNJOFE VTJOH 1BSBMMFMPHSBN USJHPOPNFUSZ BOE UIF UBTL JT MFGU UP UIF SFBEFS 4JODF UIF TIBQF PG B QBSBMMFMPHSBN XJUI TJEFT JO UIF "OPUIFS JOUFSFTUJOH QSPQFSUZ PG CPUI UIF HPMEFO SBUJP PG QIJ JT WBSJBCMF JU TFFNFE OBUVSBM GSPN UIF SFDUBOHMF BOE HPMEFO SIPNCVT JO UIJT BGPSFNFOUJPOFE UP EFëOF B AHPMEFO QBSBMMFMPHSBN DPOëHVSBUJPO JT UIBU UBO &,' = UBO #$% = ̸ ̸ BT B QBSBMMFMPHSBN "#$% XJUI JUT TJEFT BOE 0OF XBZ PG FBTJMZ FTUBCMJTIJOH UIJT JT CZ BQQMZJOH EJBHPOBMT JO UIF SBUJP QIJ FH "% = #% = φ BT UIF EPVCMF BOHMF UBO GPSNVMB CVU UIJT JT BMTP MFGU "# "$ TIPXO JO 'JHVSF &YQFSJNFOUBMMZ ESBHHJOH B BT BO FYFSDJTF UP UIF SFBEFS UP WFSJGZ EZOBNJDBMMZ DPOTUSVDUFE HFOFSBM QBSBMMFMPHSBN 4JODF EFëOJUJPOT JO NBUIFNBUJDT BSF UP TPNF VOUJM JUT TJEFT BOE EJBHPOBMT XFSF BQQSPYJNBUFMZ JO FYUFOU BSCJUSBSZ BOE UIFSF JT OP QTZDIPMPHJDBM UIF HPMEFO SBUJP HBWF B NFBTVSFNFOU GPS ̸ "#$ PG SFBTPO UP QSFGFS UIF POF UP UIF PUIFS GSPN B WJTVBM BQQSPYJNBUFMZ ◦
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At Right Angles | Vol. 6, No. 1, March 2017 67 'JHVSF $POTUSVDUJPO PG HPMEFO QBSBMMFMPHSBNT CZ TVCEJWJTJPO
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MICHAEL DE VILLIERS has worked as researcher, mathematics and science teacher at institutions across the world. Since 1991 he has been part of the University of Durban-Westville (now University of KwaZulu- Natal). He was editor of Pythagoras, the research journal of the Association of Mathematics Education of South Africa, and has been vice-chair of the SA Mathematics Olympiad since 1997. His main research interests are Geometry, Proof, Applications and Modeling, Problem Solving, and the History of Mathematics. His home page is http://dynamicmathematics-learning.com/homepage4.html. He maintains a web page for dynamic geometry sketches at http://dynamic-mathematicslearning.com/JavaGSPLinks.htm. He may be contacted on [email protected].
At Right Angles | Vol. 6, No. 1, March 2017 69 An example of constructive defining: From a GOLDEN TechSpace RECTANGLE to GOLDEN QUADRILATERALS and Beyond Part 2
MICHAEL DE VILLIERS Tis article continues the investigation started by the author in the March 2017 issue of At Right Angles, available at: http://teachersofndia.org/en/ ebook/golden-rectangle-golden-quadrilaterals-and-beyond-1 Te focus of the paper is on constructively defning various golden quadrilaterals analogously to the famous golden rectangle so that they exhibit some aspects of the golden ratio phi. Constructive defning refers to the defning of new objects by modifying or extending known defnitions or properties of existing objects. In the frst part of the paper in De Villiers (2017), diferent possible defnitions were proposed for the golden rectangle, golden rhombus and golden parallelogram, and they were compared in terms of their properties as well as ‘visual appeal’.
In this part of the paper, we shall frst look at possible defnitions for a golden isosceles trapezium as well as a golden kite, and later, at a possible defnition for a golden hexagon.
Constructively Defning a ‘Golden Isosceles Trapezium’ How can we constructively defne a ‘golden’ isosceles trapezium? Again, there are several possible options. It seems natural though, to frst consider constructing a golden isosceles trapezium ABCD in two diferent ways from a golden parallelogram (ABXD in the 1st case, and AXCD in
Keywords: Golden ratio, golden isosceles trapezium, golden kite, golden hexagon, golden triangle, golden rectangle, golden parallelogram, golden rhombus, constructive defning
74 At Right Angles | Vol. 6, No. 2, August 2017
Figure 8. Constructing a golden isosceles trapezium in two ways the 2nd case) with an acute angle of 60° as shown trapezium are also in the ratio (phi + 1), and is in Figure 8. In the frst construction shown, this therefore in the shape of the second type in Figure amounts to defning a golden isosceles trapezium 8. Te rhombus formed by the midpoints of the as an isosceles trapezium ABCD with AD // BC, sides of the frst golden isosceles trapezium is also angle ABC = 60°, and the (shorter parallel) side not any of the previously defned ‘golden’ rhombi. AD and ‘leg’ AB in the golden ratio phi. With reference to the frst construction, we could From the frst construction, it follows that triangle defne the golden isosceles trapezium without any DXC is equilateral, and therefore XC = a. Hence, reference to the 60° angle as an isosceles trapezium BC/AD = (phi + 1)/phi, which is well known to ABCD with AD // BC, and AD/AB = phi = BC/AD also equal phi1 . Tis result together with the as shown in Figure 9. similarity of isosceles triangles AED and CEB, further implies that CE/EA = BE/ED = phi. In other words, not only are the parallel sides in the golden ratio, but the diagonals also divide each other in the golden ratio. Quite nice! In the second case, however, AD/BC = phi/ (phi – 1) = (phi + 1) = phi squared. Also note in the second case, in contrast to the frst, it is the longer parallel side AD that is in the golden ratio Figure 9. Alternative defnition for frst golden to the ‘leg’ AB, and the ‘leg’ AB is in the golden isosceles trapezium ratio with the shorter side BC. So the sides of this golden isosceles trapezium form a geometric However, this is clearly not as convenient a progression from the shortest to the longest side, defnition, as such a choice of defnition requires which is quite nice too! again the use of the cosine formula to show that it implies that angle ABC = 60° (left to Subdividing the golden isosceles trapezium in the the reader to verify). As seen earlier, stating frst case in Figure 8, like the golden parallelogram one of the angles and an appropriate golden in Figure 5, by respectively constructing a ratio of sides or diagonals in the defnition, rhombus or two equilateral triangles at the ends, substantially simplifes the deductive structure. clearly does not produce an isosceles trapezium Tis illustrates the important educational point similar to the original. In this case the parallel that, generally, we choose our mathematical sides (longest/shortest) of the obtained isosceles defnitions for convenience and one of the criteria 1
1 Keep in mind that phi is defned as the solution to the quadratic equation phi2 – phi – 1 = 0. From this, it follows that phi = (phi + 1)/phi, phi = 1/(phi -1), phi/(phi +1) = phi + 1, or phi2 = phi + 1.
At Right Angles | Vol. 6, No. 2, August 2017 75 golden ratio of sides or diagonals in the definition, substantially simplifies the deductive structure.golden This ratio illustrates of sides orthe diagonals important in educationalthe definition, point substantially that, generally, simplifies we thechoose deductive our structure. This illustrates the important educational point that, generally, we choose our mathematical definitions for conveniencefor ‘convenience’ andis the oneease byof which the criteriathe other for ‘convenience’A fourth way to deisf nethe and conceptualize a golden ease mathematicalby which the othedefinitionsr properties propertiesfor convenience can can be be derived derived and from fromone it. of it. the A fourthcriteria way for isoscelesto‘convenience’ define trapezium and conceptualize couldis the be to start aagain golden with isosceles trapezium could be to start a golden triangle ABX, but this time to translate ease by which the other properties can be derived from againit. with a goldenit with thetriangle vector BXABX along, but its this‘base’ time to produce to translate it with the vector BX along its a golden isosceles trapezium ABCD as shown in ‘base’ to produceFigure a golden11. In this isosceles case, since trapezium the fgure is ABCDmade as shown in Figure 11. In this case, since the figureup is of made3 congruent up of golden 3 congruent triangles, itgolden follows thattriangles, it follows that AB/AD = phi, AB/AD = phi, and BC = 2AD (and therefore its FigureFigure 13: Fir13. stFirst case case of of golden golden kite kite and BC = 2ADdiagonals (and therefore also divide its each diagonals other in the als ratioo divide each other in the ratio 2 to 1). 2 to 1). Though one could maybe argue that the first case of a golden isosceles trapezium A fourth way to define and conceptualize Ta oughgolden one isoscelescould maybe trapezium argue that the could frst casebe to start in Figure 8 is too ‘broad’ and the one in Figure 11 is too ‘tall’ to be visually appealing, again with a golden triangle ABX, but this oftime a golden to translate isosceles trapeziumit with the in Figure vector 8 isBX too along its ‘broad’ and the one in Figure 11 is too ‘tall’ to be Figure 10. Tird golden isosceles trapeziumthere is little visually appealing, different there betwe is littleen visually the di oneferent in Figure 10 and the second case in Figure‘base’ 10: toThird produce golden a golden isosceles isosceles trapezium trapezium ABCD as shown in Figure 11. In this case, Figure 8. However,between all the four one incases Figure or 10 types and the have second interesting case mathematical properties, and AnotherFiguresince completelythe 10: figure Third di fisgoldenerent made way isoscelesup to deoff ne3 congruentand trapezium in golden Figure 8. triangles, However, allit fourfollows cases orthat types AB have/AD = phi, Another completely different way to define and conceptualizedeserve to be a known. interesting golden mathematical isosceles properties, and deserve conceptualizeand BC = a2 goldenAD (and isosceles therefore trapezium its isdiagonals to also divide each other in the ratio 2 to 1). Another completely differentagain use way a golden to triangle. define As andshown conceptualizein Figure to a be golden known. isosceles trapezium is to again use a golden triangle. As shown in Figure 10, by reflecting a golden FigureFigure 14: Second 14. Second case case of of goldengolden kite kite 10, by refectingThough a golden one triangle could ABC maybe in the argue that the first case of a golden isosceles trapezium trapezium is to again use aperpendicular golden triangle. bisector ofAs one shown of its ‘legs’ in Figure BC, 10, by reflecting a golden triangle ABC in the perpendicular bisector of one of its ‘legs’ BC, produces a ‘golden ConstructivelyAnother way Demightfning be a again‘Golden to Kite’define the pairs one,of angles and is intherefore the golden perhaps kite a littleto be more in the visually producesin Figure a ‘golden 8 is isosceles too ‘broad’ trapezium’ and where the onethe in Figure 11 is too ‘tall’ to be visually appealing, triangle ABC in the perpendicular bisector of one of its ‘legs’ BC, produces a ‘golden pleasing. Tis observation, of course, also relates isosceles trapezium’ where theratio ratioBC/AB BCis phi,/AB and is the phi, acute and ‘base’ theangle acute is ‘base’ angle is 72°. Againgolden there ratio are several as shown possible in ways Figure in which 14. Determining x from this geometric progression, there is little visually different between the one in Figure 10 and the second case in to constructively defne the concept of a ‘golden to the ratio of the diagonals, which in the frst isosceles trapezium’ where72°. theMoreover, ratio sinceBC /angleAB isBAD phi, = 108° and and the angle acute ‘base’ angle is 72°. 360 kite’.rounded An easy off way to oftwo constructing decimals, (andgives: de fxning) case 45 is .2.4084 . (roundedOf special of interest to 2 decimals) is that whilethe in Moreover, since angle BAD =ADB Figure108° = 36°, and 8. it followsHowever, angle that ADC angle all four =ABD 36°, casesis also it follows36°.or types thathave angle interesting ABD ismathematical properties, and 2 1 2 the case in Figure 14, it is 1.84 (rounded of to Moreover, since angle BADHence, = 108°AD = ABand (= DCangle), and ADC therefore = 36°, the two it follows that angle ABD is one might be to again start with a golden triangle 2 decimals), and hence the latter is closer to the also 36°. Hence, AD = AB (=parallel deserveDC ),sides and to are be thereforealso known. in the golden the tworatio, andparallel as sides are also in the andangles construct at B anand equilateral D work triangleout to beon preciselyits base as equal to 120°. This golden kite looks a little also 36°. Hence, AD = AB (= DC), and therefore the two parallel sides are also in the golden ratio phi. with the preceding case, the diagonals therefore shown in Figure 13. Since AB/BD = phi, it follows golden ratio, and as with the preceding case, the diagonals therefore also divide each ‘fatter’ than the precedingBD BC convex one, and is therefore perhaps a little more visually golden ratio, and as withalso the divide preceding each other case, into the the golden diagonals ratio. Of therefore also divide each immediately that since = by construction, To defne a golden kite that is hopefully even Figure 12: Golden isosceles trapezia of type 3 AB BC interest also, is to note that the diagonals AC and pleasi to ng. is alsoThis in observation, the golden ratio. of course, Notice thatalso relatesmore to the visually ratio appealing of the diagonals, than the previous which intwo, I other into the golden ratio. Of interest also is to note that the 2diagonals AC and DB each the same construction applies to the concave case, other into the golden ratio.DB Of each interest respectively also bisect is to the note ‘base’ that angles the at diagonalsC AC and DB each the first case is 2.40 (rounded off to 2 decimals) nextwhile thought in the ofcase def inning Figure a ‘golden 14, itkite’ is 1.84 as shown 2 but is probably not as ‘visually pleasing’ as the 3 respectively bisect the ‘base’ andangles B. at 2B and C. One more argument towards perhaps slightly favoring the golden isosceles trapezium, in Figure 15, namely, as a (convex ) kite with respectively bisect the ‘base’ angles at B and C. convex(rounded case. off to 2 decimals), and hence the latter isboth closer its sides to the and golden diagonals ratio in phi. the golden ratio. defined and constructed in Figure 9, might be that it appears in both the regular convex Another wayTo mightdefine be a againgolden to dekitefne that the pairsis hopefully even more visually appealing than the
pentagon as well as the regular star pentagon as illustrated in Figure 12. of angles in the golden kite to be in the golden previous two, I next thought of defining a ‘golden kite’ as shown in Figure 15, namely, as ratio as shown in Figure 14. Determining x from Constructively Defining a ‘Golden Kite’ thisa (convexgeometric3) progression,kite with both rounded its sides of toand two diagonals in the golden ratio. Figure 12. Golden isosceles trapezia of type 3 decimals, gives: FigureAgain 12: there Golden are isoscelesseveral possible trapezia ways of type in which3 to constructively define the concept of a One more argument towards perhaps slightly 360° x = = 45.84° One more argument towards‘golden perhaps kite’. slightly An easy favorin way gof the constructing golden isosceles (and defining) trapezium, one might be to again start with 2 favoring the golden isosceles trapezium, defned 1+φ + 2φ defined and constructed in Figurea golden 9, trianglemightand be andconstructed that construct it appears in Figure an in equilateral10, both might the be thatregulartriangle it convexon its base as shown in Figure 13. appears in both the regular convex pentagon as Of special interest is that the angles atFi Bgure and 15:D Third case: Golden kite with sides and diagonals in the golden ratio Figure 11: A fourth golden isosceles trapezium well as the regular star pentagon as illustrated in Figure 15. Tird case: Golden kite with sides and Figurepentagon 11:Figure A 11.fourth asA fourth well golden golden as the isosceles isosceles regular trapeziumSince startrapezium AB pentagon/BD = phi, as illustratedit follows immediatelyin Figure 12. that since BD = BC by construction, ABwork to out BC to be precisely equal to 120°. Tis golden �� diagonals in the golden ratio 1 Figure 12. kite looks a little1 ‘fatter’ than theThough preceding one convex can drag a dynamically constructed kite in dynamic geometry with sides is also in the golden ratio. Notice that the same construction applies to the concave3 For case, the sake of brevity we shall disregard the concave case here. Constructively Defining a ‘Golden Kite’ constructed in the golden ratio so that its diagonals are approximately also in the golden 2 In De Villiers (2009, p. 154-155; 207) a general isoscelesbut trapezium is probably with three adjnotacent as sides ‘visually equal is called apleasing’ trilateral trapezium, as andthe the convex property case. thatAgain a pair of adjacent,there congruentare several angles are bisectedpossible by the diagonalsways is inalso mentioned.which Alsoto see:constructively http://dynamicmathematicslearning.com/quad- define the concept of a ratio, making an accurate construction required the calculation of one of the angles. At tree-new-web.html 3 For the sake of brevity we shall disregard the concave case here. 2 2 ‘golden kite’. An easy way of constructing (and defining) one might be to again start with first I again tried to use the cosine rule, since it had proved effective in the case of one In De In Villiers De Villiers (2009, (2009, p. 154 p.- 155;154- 155;207) 207) a general a general isosceles isosceles trapezium trapezium with with three three adjacent adjacent sides sides equal equal is is calledcalled a trila ateral trila teraltrapezium trapezium, and76, theand property atheAt golden Right property Angles that triangle thata| Vol.pair a 6, pair No.ofand adjacent,2, of August constructadjacent, 2017 congruent congruent an equilateral angles angles are are trianglebisected bisected byon by the its the base as shown in Figure 13. golden parallelogram as well asAt Right one Angles isosceles | Vol. 6, trapezium No. 2, August case,2017 but77 with no success. diagonalsdiagonals is also is mentioned.also mentioned. Also Also see: see:http://dynamicmathematicslearning.com/quad http://dynamicmathematicslearning.com/quad-tree-tree-new-new-web.html-web.html Eventually switching strategies, and assuming AB = 1, applying the theorem of Since AB/BD = phi, it follows immediately that since BD = BC by construction, AB to BC Pythagoras to the right triangles ABE and ADE gave the following: is also in the golden ratio. Notice that the same construction applies to the concave case, BD 2 At Right Angles | Vol. 6, No. 2, August 2017 PB y2 1 but is probably not as ‘visually pleasing’ as the convex case. 4 2 BD 2 (BD y)2 2 . 4 2 Solving for y in the first equation and substituting into the second one gave the following equation in terms of BD: BD 2 BD 2 2BD 1 1 0. 4 2 This is a complex function involving both a quadratic function as well as a square root function of BD. In order to solve this equation, the easiest way as shown in Figure 16 was to use my dynamic geometry software (Sketchpad) to quickly graph the function and find the solution for x = BD = 2.20 (rounded off to 2 decimals). From there one could easily use Pythagoras to determine BE, and use the trigonometric ratios to find all the angles, giving, for example, angle BAD = 112.28°. So as expected, this golden kite is slightly ‘fatter’ and more evenly proportionate than the previous two cases. One could therefore argue that it might be visually more pleasing also. golden ratio of sides or diagonals in the definition, substantially simplifies the deductive structure.golden This ratio illustrates of sides orthe diagonals important in educationalthe definition, point substantially that, generally, simplifies we thechoose deductive our structure. This illustrates the important educational point that, generally, we choose our mathematical definitions for conveniencefor ‘convenience’ andis the oneease byof which the criteriathe other for ‘convenience’A fourth way to deisf nethe and conceptualize a golden ease mathematicalby which the othedefinitionsr properties propertiesfor convenience can can be be derived derived and from fromone it. of it. the A fourthcriteria way for isoscelesto‘convenience’ define trapezium and conceptualize couldis the be to start aagain golden with isosceles trapezium could be to start a golden triangle ABX, but this time to translate ease by which the other properties can be derived from againit. with a goldenit with thetriangle vector BXABX along, but its this‘base’ time to produce to translate it with the vector BX along its a golden isosceles trapezium ABCD as shown in ‘base’ to produceFigure a golden11. In this isosceles case, since trapezium the fgure is ABCDmade as shown in Figure 11. In this case, since the figureup is of made3 congruent up of golden 3 congruent triangles, itgolden follows thattriangles, it follows that AB/AD = phi, AB/AD = phi, and BC = 2AD (and therefore its FigureFigure 13: Fir13. stFirst case case of of golden golden kite kite and BC = 2ADdiagonals (and therefore also divide its each diagonals other in the als ratioo divide each other in the ratio 2 to 1). 2 to 1). Though one could maybe argue that the first case of a golden isosceles trapezium A fourth way to define and conceptualize Ta oughgolden one isoscelescould maybe trapezium argue that the could frst casebe to start in Figure 8 is too ‘broad’ and the one in Figure 11 is too ‘tall’ to be visually appealing, again with a golden triangle ABX, but this oftime a golden to translate isosceles trapeziumit with the in Figure vector 8 isBX too along its ‘broad’ and the one in Figure 11 is too ‘tall’ to be Figure 10. Tird golden isosceles trapeziumthere is little visually appealing, different there betwe is littleen visually the di oneferent in Figure 10 and the second case in Figure‘base’ 10: toThird produce golden a golden isosceles isosceles trapezium trapezium ABCD as shown in Figure 11. In this case, Figure 8. However,between all the four one incases Figure or 10 types and the have second interesting case mathematical properties, and AnotherFiguresince completelythe 10: figure Third di fisgoldenerent made way isoscelesup to deoff ne3 congruentand trapezium in golden Figure 8. triangles, However, allit fourfollows cases orthat types AB have/AD = phi, Another completely different way to define and conceptualizedeserve to be a known. interesting golden mathematical isosceles properties, and deserve conceptualizeand BC = a2 goldenAD (and isosceles therefore trapezium its isdiagonals to also divide each other in the ratio 2 to 1). Another completely differentagain use way a golden to triangle. define As andshown conceptualizein Figure to a be golden known. isosceles trapezium is to again use a golden triangle. As shown in Figure 10, by reflecting a golden FigureFigure 14: Second 14. Second case case of of goldengolden kite kite 10, by refectingThough a golden one triangle could ABC maybe in the argue that the first case of a golden isosceles trapezium trapezium is to again use aperpendicular golden triangle. bisector ofAs one shown of its ‘legs’ in Figure BC, 10, by reflecting a golden triangle ABC in the perpendicular bisector of one of its ‘legs’ BC, produces a ‘golden ConstructivelyAnother way Demightfning be a again‘Golden to Kite’define the pairs one,of angles and is intherefore the golden perhaps kite a littleto be more in the visually producesin Figure a ‘golden 8 is isosceles too ‘broad’ trapezium’ and where the onethe in Figure 11 is too ‘tall’ to be visually appealing, triangle ABC in the perpendicular bisector of one of its ‘legs’ BC, produces a ‘golden pleasing. Tis observation, of course, also relates isosceles trapezium’ where theratio ratioBC/AB BCis phi,/AB and is the phi, acute and ‘base’ theangle acute is ‘base’ angle is 72°. Againgolden there ratio are several as shown possible in ways Figure in which 14. Determining x from this geometric progression, there is little visually different between the one in Figure 10 and the second case in to constructively defne the concept of a ‘golden to the ratio of the diagonals, which in the frst isosceles trapezium’ where72°. theMoreover, ratio sinceBC /angleAB isBAD phi, = 108° and and the angle acute ‘base’ angle is 72°. 360 kite’.rounded An easy off way to oftwo constructing decimals, (andgives: de fxning) case 45 is .2.4084 . (roundedOf special of interest to 2 decimals) is that whilethe in Moreover, since angle BAD =ADB Figure108° = 36°, and 8. it followsHowever, angle that ADC angle all four =ABD 36°, casesis also it follows36°.or types thathave angle interesting ABD ismathematical properties, and 2 1 2 the case in Figure 14, it is 1.84 (rounded of to Moreover, since angle BADHence, = 108°AD = ABand (= DCangle), and ADC therefore = 36°, the two it follows that angle ABD is one might be to again start with a golden triangle 2 decimals), and hence the latter is closer to the also 36°. Hence, AD = AB (=parallel deserveDC ),sides and to are be thereforealso known. in the golden the tworatio, andparallel as sides are also in the andangles construct at B anand equilateral D work triangleout to beon preciselyits base as equal to 120°. This golden kite looks a little also 36°. Hence, AD = AB (= DC), and therefore the two parallel sides are also in the golden ratio phi. with the preceding case, the diagonals therefore shown in Figure 13. Since AB/BD = phi, it follows golden ratio, and as with the preceding case, the diagonals therefore also divide each ‘fatter’ than the precedingBD BC convex one, and is therefore perhaps a little more visually golden ratio, and as withalso the divide preceding each other case, into the the golden diagonals ratio. Of therefore also divide each immediately that since = by construction, To defne a golden kite that is hopefully even Figure 12: Golden isosceles trapezia of type 3 AB BC interest also, is to note that the diagonals AC and pleasi to ng. is alsoThis in observation, the golden ratio. of course, Notice thatalso relatesmore to the visually ratio appealing of the diagonals, than the previous which intwo, I other into the golden ratio. Of interest also is to note that the 2diagonals AC and DB each the same construction applies to the concave case, other into the golden ratio.DB Of each interest respectively also bisect is to the note ‘base’ that angles the at diagonalsC AC and DB each the first case is 2.40 (rounded off to 2 decimals) nextwhile thought in the ofcase def inning Figure a ‘golden 14, itkite’ is 1.84 as shown 2 but is probably not as ‘visually pleasing’ as the 3 respectively bisect the ‘base’ andangles B. at 2B and C. One more argument towards perhaps slightly favoring the golden isosceles trapezium, in Figure 15, namely, as a (convex ) kite with respectively bisect the ‘base’ angles at B and C. convex(rounded case. off to 2 decimals), and hence the latter isboth closer its sides to the and golden diagonals ratio in phi. the golden ratio. defined and constructed in Figure 9, might be that it appears in both the regular convex Another wayTo mightdefine be a againgolden to dekitefne that the pairsis hopefully even more visually appealing than the pentagon as well as the regular star pentagon as illustrated in Figure 12. of angles in the golden kite to be in the golden previous two, I next thought of defining a ‘golden kite’ as shown in Figure 15, namely, as ratio as shown in Figure 14. Determining x from Constructively Defining a ‘Golden Kite’ thisa (convexgeometric3) progression,kite with both rounded its sides of toand two diagonals in the golden ratio. Figure 12. Golden isosceles trapezia of type 3 decimals, gives: FigureAgain 12: there Golden are isoscelesseveral possible trapezia ways of type in which3 to constructively define the concept of a One more argument towards perhaps slightly 360° x = = 45.84° One more argument towards‘golden perhaps kite’. slightly An easy favorin way gof the constructing golden isosceles (and defining) trapezium, one might be to again start with 2 favoring the golden isosceles trapezium, defned 1+φ + 2φ defined and constructed in Figurea golden 9, trianglemightand be andconstructed that construct it appears in Figure an in equilateral10, both might the be thatregulartriangle it convexon its base as shown in Figure 13. appears in both the regular convex pentagon as Of special interest is that the angles atFi Bgure and 15:D Third case: Golden kite with sides and diagonals in the golden ratio Figure 11: A fourth golden isosceles trapezium well as the regular star pentagon as illustrated in Figure 15. Tird case: Golden kite with sides and Figurepentagon 11:Figure A 11.fourth asA fourth well golden golden as the isosceles isosceles regular trapeziumSince startrapezium AB pentagon/BD = phi, as illustratedit follows immediatelyin Figure 12. that since BD = BC by construction, ABwork to out BC to be precisely equal to 120°. Tis golden �� diagonals in the golden ratio 1 Figure 12. kite looks a little1 ‘fatter’ than theThough preceding one convex can drag a dynamically constructed kite in dynamic geometry with sides is also in the golden ratio. Notice that the same construction applies to the concave3 For case, the sake of brevity we shall disregard the concave case here. Constructively Defining a ‘Golden Kite’ constructed in the golden ratio so that its diagonals are approximately also in the golden 2 In De Villiers (2009, p. 154-155; 207) a general isoscelesbut trapezium is probably with three adjnotacent as sides ‘visually equal is called apleasing’ trilateral trapezium, as andthe the convex property case. thatAgain a pair of adjacent,there congruentare several angles are bisectedpossible by the diagonalsways is inalso mentioned.which Alsoto see:constructively http://dynamicmathematicslearning.com/quad- define the concept of a ratio, making an accurate construction required the calculation of one of the angles. At tree-new-web.html 3 For the sake of brevity we shall disregard the concave case here. 2 2 ‘golden kite’. An easy way of constructing (and defining) one might be to again start with first I again tried to use the cosine rule, since it had proved effective in the case of one In De In Villiers De Villiers (2009, (2009, p. 154 p.- 155;154- 155;207) 207) a general a general isosceles isosceles trapezium trapezium with with three three adjacent adjacent sides sides equal equal is is calledcalled a trila ateral trila teraltrapezium trapezium, and76, theand property atheAt golden Right property Angles that triangle thata| Vol.pair a 6, pair No.ofand adjacent,2, of August constructadjacent, 2017 congruent congruent an equilateral angles angles are are trianglebisected bisected byon by the its the base as shown in Figure 13. golden parallelogram as well asAt Right one Angles isosceles | Vol. 6, trapezium No. 2, August case,2017 but77 with no success. diagonalsdiagonals is also is mentioned.also mentioned. Also Also see: see:http://dynamicmathematicslearning.com/quad http://dynamicmathematicslearning.com/quad-tree-tree-new-new-web.html-web.html Eventually switching strategies, and assuming AB = 1, applying the theorem of Since AB/BD = phi, it follows immediately that since BD = BC by construction, AB to BC Pythagoras to the right triangles ABE and ADE gave the following: is also in the golden ratio. Notice that the same construction applies to the concave case, BD 2 At Right Angles | Vol. 6, No. 2, August 2017 PB y2 1 but is probably not as ‘visually pleasing’ as the convex case. 4 2 BD 2 (BD y)2 2 . 4 2 Solving for y in the first equation and substituting into the second one gave the following equation in terms of BD: BD 2 BD 2 2BD 1 1 0. 4 2 This is a complex function involving both a quadratic function as well as a square root function of BD. In order to solve this equation, the easiest way as shown in Figure 16 was to use my dynamic geometry software (Sketchpad) to quickly graph the function and find the solution for x = BD = 2.20 (rounded off to 2 decimals). From there one could easily use Pythagoras to determine BE, and use the trigonometric ratios to find all the angles, giving, for example, angle BAD = 112.28°. So as expected, this golden kite is slightly ‘fatter’ and more evenly proportionate than the previous two cases. One could therefore argue that it might be visually more pleasing also. Tough one can drag a dynamically constructed quickly graph the function and fnd the solution cases since the angles only difer by a few degrees decimals), and since it is further from the golden kite in dynamic geometry with sides constructed for x = BD = 2.20 (rounded of to 2 decimals). (as can be easily verifed by calculation by the ratio, explains the elongated, thinner shape in in the golden ratio so that its diagonals are From there one could easily use Pythagoras to reader). Also note that for the construction in comparison with the golden kites in Figures 14 approximately also in the golden ratio, making determine BE, and use the trigonometric ratios Figure 17, as we’ve already seen earlier, AD to AB and 15. B C + D B + C + D we’ve already seen earlier, AD to AB will be in the golden ratio, if and only if, isosceles an"' accurate= D construction− = required− the calculation= T B; to fnd all the angles, giving, for example, angle will be in the golden ratio, if and only if, isosceles of one of− the angles. At frst I again tried to− use BAD = 112.28°. So as expected, this golden kite trapezium ABCD is a goldentrapezium rectangle. ABCD is a golden rectangle. Last, but not least, one can also choose to defne the cosine rule, since it had proved efective in is slightly ‘fatter’ and more evenly proportionate the famous Penrose kite and dart as ‘golden kites’, B + C + D B + C D the$& case= C of one− golden parallelogram= − as= wellT asD , than the previous two cases. One could therefore which are illustrated in Figure 19. As can be seen, − B C + D B + C + D − one"' isosceles= D trapezium− = case,− but with no= Tsuccess.B; argue that it might be visually more pleasing also. they can be obtained from a rhombus with angles − − of 72° and 108° by dividing the long diagonal Eventually switchingB C + strategies,D B + Cand Dassuming AB = $%1, applying= B the− theorem= of Pythagoras− = Tto theD; In addition, the midpoint rectangle of the third of the rhombus in the ratio of phi so that the − B +C + D B +C D − right$& = trianglesC − ABE and ADE= gave −the following:= T D, golden kite in Figure 15, since its diagonals are in ‘symmetrical’ diagonal of the Penrose kite is in − − the golden ratio, is a golden rectangle. the ratio phi to the ‘symmetrical’ diagonal of the #% dart. It is left to the reader to verify that from B CZ+ +D B +=C D On that note, jumping back to the previous $% = B − =φ − = T D; this construction it follows that both the Penrose − − section, this reminded me that a ffth way in kite and dart have their sides in the ratio of phi. #% which we could defne a golden isosceles trapezoid +(#% Z) = φ . Moreover, the Penrose kites and darts can be used − might be to defne it as an isosceles trapezium φ #% to tile the plane non-periodically, and the ratio of Z + = with its mid-segments KM and LN in the golden φ the number of kites to darts tends towards phi as Solving for y in the frst equation and substituting ratio as shown in Figure 17, since its midpoint #% the number of tiles increase (Darvas, 2007: 204). into#% the second#%#% one gave the followingφ + equation= . rhombus would then be a golden rhombus (with − +(#%− Z) −= φ . Of additional interest, is that the ‘fat’ rhombus ! −φ in terms of BDφ : diagonals in golden ratio). However, in general, formed by the Penrose kite and dart as shown such an isosceles trapezium is dynamic and can 6 = 6 + S (U U ), in Figure 19, also non-periodically tiles with 6 change shape, and we need to add a further #% − the ‘thin’ rhombus given earlier by the second #% #% φ + = . property to fx its shape. For example, in the 1st − ! − φ − Figure 18: Constructing golden kite tangent to circumcircle of KLMNgolden rhombus in Figure 7, and the ratio of the 'MFB case shown in Figure 17 we could impose the Figure 18. Constructing golden kite tangent to MJN = . nd circumcircle of KLMN number of ‘fat’ rhombi to ‘thin’ rhombi similarly Tis is a complexI function I involving both a condition that BC/AD = phi, or as in the 2 case, Since all isosceles trapezia are cyclic (and all kites are circumscribed), another way to 6 = →6 + S6(U U), tends towards phi as the number of tiles increase quadratic function as well as a− square root we can have AB = AD = DC (so the base angles conceptualize and constructivelySince all isosceles define trapezia a ‘golde are cyclicn kite’ (and would all kites be to also(Darvas, construct 2007: the 202). Te interested reader will function of BDΔ.G To= IGsolve′(Y )+this 'MFBequation,, the at B and C would respectivelyFigure be 16: bisected Solving by thefor BD by graphing are circumscribed), another way to conceptualize 'MFB fnd various websites on the Internet giving easiest way as shownMJN in Figure= .16 was to use diagonals DB and AC). As can be seen, it is very ‘dual’ of each of the goldenand constructively isosceles trapezia defne a ‘goldenalready kite’ discussed. would For example,examples consider of Penrose tiles of kites and darts as well my dynamic geometryI Isoftware (Sketchpad) to In di addition,fcult to visually the midpoint distinguish rectangle between of these the two third golden kite in Figure 15, since its → the golden isosceles trapeziumbe to also KLMNconstruct defined the ‘dual’ in Figure of each 10, of theand its circumcircleas of the as mentioned shown rhombi. diagonals are in the golden ratio, is a golden rectangle. golden isosceles trapezia already discussed. For ΔG = IG ′(Y)+'MFB, in Figure 18. As was example,the case considerfor the gothelden golden rectangle, isosceles wetrapezium can now similarly construct perpendiculars to the radiiKLMN at eachdefned of inthe Figure vertices 10, toand produce its circumcircle a corresponding dual ‘golden as shown in Figure 18. As was the case for the Figure 16: Solving for BD by graphing kite’ ABCD. It is nowgolden left to rectangle, the reader we canto verify now similarly that CBD construct is a golden triangle (hence BC/BD = phi) and angleperpendiculars ABC = angle to theBAD radii = angleat each ADC of the = vertices 108°. In addition ABCD has In addition, the midpoint rectangle of the third golden kite in Figure 15, since its to produce a corresponding dual ‘golden kite’ the dual property (to theABCD angle. It bisection is now left of to two the anglesreader toby verify diagonals that in KLMN) of K and N diagonals are in the golden ratio, is a golden rectangle. being respective midpointsCBD is ofa goldenAB and triangle AD4 .(hence The readerBC/BD may= phi) also wish to verify that Figure 17: Fifth case: Golden isosceles trapezia via midsegments in goldenand ratio angle ABC = angle BAD = angle ADC = 108°. AC/BD = 1.90 (roundedIn additionoff to 2 ABCDdecimals), has the and dual since property it is further(to the from the golden ratio, On that note, jumping back to the previous section, this reminded me that a fifth way in explains the elongated,angle thinner bisection shape of intwo comparison angles by diagonals with the in golden kites in Figures 14 KLMN) of K and N being respective midpoints which we could define a golden isosceles trapezoid might be to define it as an isosceles FigureFigure 19: 19. Penrose Penrose kite kite and and dart dart and 15. of AB and AD . Te reader may also wish to trapezium with its mid-segments KM and LN in the golden ratio as shown in Figureverify that 17, AC/BD 1 = 1.90 (rounded of to 2 Last, but not least, one can also choose to define the famous Penrose kite and dart as
since its midpoint rhombus would then be a golden rhombus4 (with diagonals in golden Figure 16: Solving for BD by graphing 4 ‘golden kites’, which are illustrated in Figure 19. As can be seen, they can be obtained Figure 17: Fifth case: Golden isosceles trapezia via midsegments in golden ratio In De Villiers (2009, p. 154 In- 155;De Villiers 207) (2009, , a general p. 154-155; kite 207) with , a threegeneral adjacentkite with three angles adjacent equal angles is calledequal is calleda a triangular kite, and the property that a pair of Figure 16. Solving for BD by graphing ratio). However,Figure 17. Fifth in general case: Golden such isosceles an isosceles trapezia trapezium istriangular dynamic kite and, and can the change propertyadjacent, shape, that congruent a pair of sides adjacent, are bisected congruent by the tangent sides points are of bisected the incircle by is thealso mentioned.tangent points Te Penrose kite in Figure 19 is also an example of a from a rhombus with angles of 72° and 108° by dividing the long diagonal of the In addition, the midpoint rectangle of the third golden kite in Figure 15, sincevia midsegments its in golden ratio of the incircle is also mentioned.st triangular The kite. Penrose Also see: kite http://dynamicmathematicslearning.com/quad-tree-new-web.html in Figure 19 is also an example of a triangular kite. Also On that note, jumping back to the andprevious we need section, to add this a furtherreminded property me that to afix fifth its wayshape. in Forsee: example, http://dynamicmathemat in the 1 caseicslearning.com/quad shown -tree-new-web.html diagonals are in the golden ratio, is a golden rectangle. rhombus in the ratio of phi so that the ‘symmetrical’ diagonal of the Penrose kite is in the nd 78 which we could define a golden isoscelesin Figure trapezoid 17 we could might impose be to define the condition it as an thatisosceles BC/AD = phi, or as in the 2 case, we 79 At Right Angles | Vol. 6, No. 2, August 2017 ratio phi to the ‘symmetrical’ diagonalAt Right Angles of the | Vol. dart. 6, No. It 2, isAugust left 2017 to the reader to verify that trapezium with its mid-segments KMcan andhave LN AB in = the AD golden = DC (soratio the as base shown angles in Figureat B and 17, C would respectively be bisected by from this construction it follows that both the Penrose kite and dart have their sides in the since its midpoint rhombus would thethen diagonals be a golden DB rhombusand AC). (with As can diagonals be seen, in itgolden is very difficult to visually distinguish ratio of phi. Moreover, the Penrose kites and darts can be used to tile the plane non- ratio). However, in general such anbetween isosceles these trapezium two cases is dynamic since the and angles can change only differshape, by a few degrees (as can be easily At Right Angles | Vol. 6, No. 2, August 2017 PB periodically, and the ratio of the number of kites to darts tends towards phi as the number and we need to add a further propertyverified to fix by its calculation shape. For byexample, the reader). in the Also 1st case note shown that for the construction in Figure 17, as of tiles increase (Darvas, 2007: 204). Of additional interest, is that the ‘fat’ rhombus Figure 17: Fifth case: Goldenin isoscelesFigure 17 trapezia we could via midsegments impose the conditionin golden ratiothat BC/AD = phi, or as in the 2nd case, we formed by the Penrose kite and dart as shown in Figure 19, also non-periodically tiles can have AB = AD = DC (so the base angles at B and C would respectively be bisected by On that note, jumping back to the previous section, this reminded me that a fifth way in with the ‘thin’ rhombus given earlier by the second golden rhombus in Figure 7, and the the diagonals DB and AC). As can be seen, it is very difficult to visually distinguish which we could define a golden isosceles trapezoid might be to define it as an isosceles ratio of the number of ‘fat’ rhombi to ‘thin’ rhombi similarly tends towards phi as the trapezium with its mid-segmentsbetween KM and theseLN in two the casesgolden since ratio theas shown angles in only Figure differ 17, by a few degrees (as can be easily number of tiles increase (Darvas, 2007: 202). The interested reader will find various since its midpoint rhombus wouldverified then beby acalculation golden rhombus by the (with reader). diagonals Also notein golden that for the construction in Figure 17, as websites on the Internet giving examples of Penrose tiles of kites and darts as well as of ratio). However, in general such an isosceles trapezium is dynamic and can change shape, the mentioned rhombi. and we need to add a further property to fix its shape. For example, in the 1st case shown in Figure 17 we could impose the condition that BC/AD = phi, or as in the 2nd case, we Constructively Defining Other ‘Golden Quadrilaterals’ can have AB = AD = DC (so the base angles at B and C would respectively be bisected by This investigation has already become longer than I’d initially anticipated, and it is time the diagonals DB and AC). As can be seen, it is very difficult to visually distinguish to finish it off before I start boring the reader. Moreover, my main objective of showing between these two cases since the angles only differ by a few degrees (as can be easily constructive defining in action has hopefully been achieved by now. verified by calculation by the reader). Also note that for the construction in Figure 17, as However, I’d like to point out that there are several other types of quadrilaterals for which one can similarly explore ways to define ‘golden quadrilaterals’, e.g., cyclic Tough one can drag a dynamically constructed quickly graph the function and fnd the solution cases since the angles only difer by a few degrees decimals), and since it is further from the golden kite in dynamic geometry with sides constructed for x = BD = 2.20 (rounded of to 2 decimals). (as can be easily verifed by calculation by the ratio, explains the elongated, thinner shape in in the golden ratio so that its diagonals are From there one could easily use Pythagoras to reader). Also note that for the construction in comparison with the golden kites in Figures 14 approximately also in the golden ratio, making determine BE, and use the trigonometric ratios Figure 17, as we’ve already seen earlier, AD to AB and 15. B C + D B + C + D we’ve already seen earlier, AD to AB will be in the golden ratio, if and only if, isosceles an"' accurate= D construction− = required− the calculation= T B; to fnd all the angles, giving, for example, angle will be in the golden ratio, if and only if, isosceles of one of− the angles. At frst I again tried to− use BAD = 112.28°. So as expected, this golden kite trapezium ABCD is a goldentrapezium rectangle. ABCD is a golden rectangle. Last, but not least, one can also choose to defne the cosine rule, since it had proved efective in is slightly ‘fatter’ and more evenly proportionate the famous Penrose kite and dart as ‘golden kites’, B + C + D B + C D the$& case= C of one− golden parallelogram= − as= wellT asD , than the previous two cases. One could therefore which are illustrated in Figure 19. As can be seen, − B C + D B + C + D − one"' isosceles= D trapezium− = case,− but with no= Tsuccess.B; argue that it might be visually more pleasing also. they can be obtained from a rhombus with angles − − of 72° and 108° by dividing the long diagonal Eventually switchingB C + strategies,D B + Cand Dassuming AB = $%1, applying= B the− theorem= of Pythagoras− = Tto theD; In addition, the midpoint rectangle of the third of the rhombus in the ratio of phi so that the − B +C + D B +C D − right$& = trianglesC − ABE and ADE= gave −the following:= T D, golden kite in Figure 15, since its diagonals are in ‘symmetrical’ diagonal of the Penrose kite is in − − the golden ratio, is a golden rectangle. the ratio phi to the ‘symmetrical’ diagonal of the #% dart. It is left to the reader to verify that from B CZ+ +D B +=C D On that note, jumping back to the previous $% = B − =φ − = T D; this construction it follows that both the Penrose − − section, this reminded me that a ffth way in kite and dart have their sides in the ratio of phi. #% which we could defne a golden isosceles trapezoid +(#% Z) = φ . Moreover, the Penrose kites and darts can be used − might be to defne it as an isosceles trapezium φ #% to tile the plane non-periodically, and the ratio of Z + = with its mid-segments KM and LN in the golden φ the number of kites to darts tends towards phi as Solving for y in the frst equation and substituting ratio as shown in Figure 17, since its midpoint #% the number of tiles increase (Darvas, 2007: 204). into#% the second#%#% one gave the followingφ + equation= . rhombus would then be a golden rhombus (with − +(#%− Z) −= φ . Of additional interest, is that the ‘fat’ rhombus ! −φ in terms of BDφ : diagonals in golden ratio). However, in general, formed by the Penrose kite and dart as shown such an isosceles trapezium is dynamic and can 6 = 6 + S (U U ), in Figure 19, also non-periodically tiles with 6 change shape, and we need to add a further #% − the ‘thin’ rhombus given earlier by the second #% #% φ + = . property to fx its shape. For example, in the 1st − ! − φ − Figure 18: Constructing golden kite tangent to circumcircle of KLMNgolden rhombus in Figure 7, and the ratio of the 'MFB case shown in Figure 17 we could impose the Figure 18. Constructing golden kite tangent to MJN = . nd circumcircle of KLMN number of ‘fat’ rhombi to ‘thin’ rhombi similarly Tis is a complexI function I involving both a condition that BC/AD = phi, or as in the 2 case, Since all isosceles trapezia are cyclic (and all kites are circumscribed), another way to 6 = →6 + S6(U U), tends towards phi as the number of tiles increase quadratic function as well as a− square root we can have AB = AD = DC (so the base angles conceptualize and constructivelySince all isosceles define trapezia a ‘golde are cyclicn kite’ (and would all kites be to also(Darvas, construct 2007: the 202). Te interested reader will function of BDΔ.G To= IGsolve′(Y )+this 'MFBequation,, the at B and C would respectivelyFigure be 16: bisected Solving by thefor BD by graphing are circumscribed), another way to conceptualize 'MFB fnd various websites on the Internet giving easiest way as shownMJN in Figure= .16 was to use diagonals DB and AC). As can be seen, it is very ‘dual’ of each of the goldenand constructively isosceles trapezia defne a ‘goldenalready kite’ discussed. would For example,examples consider of Penrose tiles of kites and darts as well my dynamic geometryI Isoftware (Sketchpad) to In di addition,fcult to visually the midpoint distinguish rectangle between of these the two third golden kite in Figure 15, since its → the golden isosceles trapeziumbe to also KLMNconstruct defined the ‘dual’ in Figure of each 10, of theand its circumcircleas of the as mentioned shown rhombi. diagonals are in the golden ratio, is a golden rectangle. golden isosceles trapezia already discussed. For ΔG = IG ′(Y)+'MFB, in Figure 18. As was example,the case considerfor the gothelden golden rectangle, isosceles wetrapezium can now similarly construct perpendiculars to the radiiKLMN at eachdefned of inthe Figure vertices 10, toand produce its circumcircle a corresponding dual ‘golden as shown in Figure 18. As was the case for the Figure 16: Solving for BD by graphing kite’ ABCD. It is nowgolden left to rectangle, the reader we canto verify now similarly that CBD construct is a golden triangle (hence BC/BD = phi) and angleperpendiculars ABC = angle to theBAD radii = angleat each ADC of the = vertices 108°. In addition ABCD has In addition, the midpoint rectangle of the third golden kite in Figure 15, since its to produce a corresponding dual ‘golden kite’ the dual property (to theABCD angle. It bisection is now left of to two the anglesreader toby verify diagonals that in KLMN) of K and N diagonals are in the golden ratio, is a golden rectangle. being respective midpointsCBD is ofa goldenAB and triangle AD4 .(hence The readerBC/BD may= phi) also wish to verify that Figure 17: Fifth case: Golden isosceles trapezia via midsegments in goldenand ratio angle ABC = angle BAD = angle ADC = 108°. AC/BD = 1.90 (roundedIn additionoff to 2 ABCDdecimals), has the and dual since property it is further(to the from the golden ratio, On that note, jumping back to the previous section, this reminded me that a fifth way in explains the elongated,angle thinner bisection shape of intwo comparison angles by diagonals with the in golden kites in Figures 14 KLMN) of K and N being respective midpoints which we could define a golden isosceles trapezoid might be to define it as an isosceles FigureFigure 19: 19. Penrose Penrose kite kite and and dart dart and 15. of AB and AD . Te reader may also wish to trapezium with its mid-segments KM and LN in the golden ratio as shown in Figureverify that 17, AC/BD 1 = 1.90 (rounded of to 2 Last, but not least, one can also choose to define the famous Penrose kite and dart as since its midpoint rhombus would then be a golden rhombus4 (with diagonals in golden Figure 16: Solving for BD by graphing 4 ‘golden kites’, which are illustrated in Figure 19. As can be seen, they can be obtained Figure 17: Fifth case: Golden isosceles trapezia via midsegments in golden ratio In De Villiers (2009, p. 154 In- 155;De Villiers 207) (2009, , a general p. 154-155; kite 207) with , a threegeneral adjacentkite with three angles adjacent equal angles is calledequal is calleda a triangular kite, and the property that a pair of Figure 16. Solving for BD by graphing ratio). However,Figure 17. Fifth in general case: Golden such isosceles an isosceles trapezia trapezium istriangular dynamic kite and, and can the change propertyadjacent, shape, that congruent a pair of sides adjacent, are bisected congruent by the tangent sides points are of bisected the incircle by is thealso mentioned.tangent points Te Penrose kite in Figure 19 is also an example of a from a rhombus with angles of 72° and 108° by dividing the long diagonal of the In addition, the midpoint rectangle of the third golden kite in Figure 15, sincevia midsegments its in golden ratio of the incircle is also mentioned.st triangular The kite. Penrose Also see: kite http://dynamicmathematicslearning.com/quad-tree-new-web.html in Figure 19 is also an example of a triangular kite. Also On that note, jumping back to the andprevious we need section, to add this a furtherreminded property me that to afix fifth its wayshape. in Forsee: example, http://dynamicmathemat in the 1 caseicslearning.com/quad shown -tree-new-web.html diagonals are in the golden ratio, is a golden rectangle. rhombus in the ratio of phi so that the ‘symmetrical’ diagonal of the Penrose kite is in the nd 78 which we could define a golden isoscelesin Figure trapezoid 17 we could might impose be to define the condition it as an thatisosceles BC/AD = phi, or as in the 2 case, we 79 At Right Angles | Vol. 6, No. 2, August 2017 ratio phi to the ‘symmetrical’ diagonalAt Right Angles of the | Vol. dart. 6, No. It 2, isAugust left 2017 to the reader to verify that trapezium with its mid-segments KMcan andhave LN AB in = the AD golden = DC (soratio the as base shown angles in Figureat B and 17, C would respectively be bisected by from this construction it follows that both the Penrose kite and dart have their sides in the since its midpoint rhombus would thethen diagonals be a golden DB rhombusand AC). (with As can diagonals be seen, in itgolden is very difficult to visually distinguish ratio of phi. Moreover, the Penrose kites and darts can be used to tile the plane non- ratio). However, in general such anbetween isosceles these trapezium two cases is dynamic since the and angles can change only differshape, by a few degrees (as can be easily At Right Angles | Vol. 6, No. 2, August 2017 PB periodically, and the ratio of the number of kites to darts tends towards phi as the number and we need to add a further propertyverified to fix by its calculation shape. For byexample, the reader). in the Also 1st case note shown that for the construction in Figure 17, as of tiles increase (Darvas, 2007: 204). Of additional interest, is that the ‘fat’ rhombus Figure 17: Fifth case: Goldenin isoscelesFigure 17 trapezia we could via midsegments impose the conditionin golden ratiothat BC/AD = phi, or as in the 2nd case, we formed by the Penrose kite and dart as shown in Figure 19, also non-periodically tiles can have AB = AD = DC (so the base angles at B and C would respectively be bisected by On that note, jumping back to the previous section, this reminded me that a fifth way in with the ‘thin’ rhombus given earlier by the second golden rhombus in Figure 7, and the the diagonals DB and AC). As can be seen, it is very difficult to visually distinguish which we could define a golden isosceles trapezoid might be to define it as an isosceles ratio of the number of ‘fat’ rhombi to ‘thin’ rhombi similarly tends towards phi as the trapezium with its mid-segmentsbetween KM and theseLN in two the casesgolden since ratio theas shown angles in only Figure differ 17, by a few degrees (as can be easily number of tiles increase (Darvas, 2007: 202). The interested reader will find various since its midpoint rhombus wouldverified then beby acalculation golden rhombus by the (with reader). diagonals Also notein golden that for the construction in Figure 17, as websites on the Internet giving examples of Penrose tiles of kites and darts as well as of ratio). However, in general such an isosceles trapezium is dynamic and can change shape, the mentioned rhombi. and we need to add a further property to fix its shape. For example, in the 1st case shown in Figure 17 we could impose the condition that BC/AD = phi, or as in the 2nd case, we Constructively Defining Other ‘Golden Quadrilaterals’ can have AB = AD = DC (so the base angles at B and C would respectively be bisected by This investigation has already become longer than I’d initially anticipated, and it is time the diagonals DB and AC). As can be seen, it is very difficult to visually distinguish to finish it off before I start boring the reader. Moreover, my main objective of showing between these two cases since the angles only differ by a few degrees (as can be easily constructive defining in action has hopefully been achieved by now. verified by calculation by the reader). Also note that for the construction in Figure 17, as However, I’d like to point out that there are several other types of quadrilaterals for which one can similarly explore ways to define ‘golden quadrilaterals’, e.g., cyclic Constructively Defning Other equi-angled, cyclic hexagon that all the pairs of compared in terms of the number of properties, investigation has hopefully also contributed a ‘Golden Quadrilaterals’ adjacent sides as shown in Figure 19 are in the ease of construction or of proof, and, in this little bit to demystifying where defnitions come Tis investigation has already become longer than golden ratio; i.e., a ‘golden (cyclic) hexagon’. It particular case in relation to the golden ratio, from, and that they don’t just pop out of the I’d initially anticipated, and it is time to fnish it is left to the reader to verify that if FA/AB = phi, perhaps also of visual appeal. Moreover, it was air into a mathematician’s mind or suddenly 7 of before I start boring the reader. Moreover, my then AL/LM = phi , etc. In other words, the main shown how some defnitions of the same object magically appear in print in a school textbook. main objective of showing constructive defning diagonals divide each other into the golden ratio. might be more convenient than others in terms of 3FGFSFODFTthe deductive derivation of other properties not In a classroom context, if a teacher were to ask in action has hopefully been achieved by now. Te observant reader would also note that ABEF, students to suggest various possible defnitions contained%F 7JMMJFST in .the defnition.4PNF "EWFOUVSFT JO &VDMJEFBO (FPNFUSZ -VMV 1SFTT However, I’d like to point out that there are ABCD and CDEF, are all three golden trapezia for golden quadrilaterals or golden hexagons several other types of quadrilaterals for which of the type constructed and defned in the frst Te&SOFTU process 1 of constructiveDzF 1IJMPTPQIZ def PGning .BUIFNBUJDT also generally &EVDBUJPO -POEPO of di 'BMNFSferent 1SFTT types, it is likely that they would case in Figure 8. Moreover, ALNF, ABCF, etc., applies to the defnition and exploration of propose several of the examples discussed here, one can similarly explore ways to defne ‘golden 'SFVEFOUIBM ) .BUIFNBUJDT BT BO &EVDBUJPOBM 5BTL % 3FJEFM %PSESFDIU )PMMBOE quadrilaterals’, e.g., cyclic quadrilaterals, are golden trapezia of the second type constructed diferent axiom systems in pure, mathematical and perhaps even a few not explored here. circumscribed quadrilaterals, trapeziums5, and defned in Figure 8. research(SPTTNBO where 1 FU quite BM often %P existing 1FPQMF 1SFGFSaxiom *SSBUJPOBM 3BUJPT "Involving /FX -PPL students BU UIF (PMEFO in an 4FDUJPO activity 4UVEFOU like SFTFBSDIthis would DPOEVDUFE JO JO UIF %FQU PG "QQMJFE $PNQVUFS 4DJFODF 6OJWFSTJUZ PG #BNCFSH "DDFTTFE PO 0DU BU 5 systems are used as starting blocks which are then not only more realistically simulate actual quadrilaterals, circumscribedbi-centric quadrilaterals, quadrilaterals, orthodiagonal trapeziums , bi-centric quadrilaterals, IUUQTXXXBDBEFNJBFEVɨF@(PMEFO@3BUJP quadrilaterals, equidiagonal quadrilaterals, etc. modifed, adapted, generalized, etc., to create mathematical research, but also provide students orthodiagonal quadrilaterals, equidiagonal quadrilaterals, etc. and-PFC explore "- new 7BSOFZ mathematical 8 %PFTtheories. UIF (PMEFO So this 4QJSBM &YJTU with BOE *G a /PU more 8IFSF personal JT JUT $FOUFS sense of *O )BSHJUUBJ ownership * over 1JDLPWFS the $" little episode4QJSBM encapsulates 4JNJMBSJUZ 4JOHBQPSF at an elementary 8PSME 4DJFOUJëD level QQ mathematical content instead of being seen as some4FSSB of .the main &EJUJPOresearch methodologies%JTDPWFSJOH (FPNFUSZ used "O *OWFTUJHBUJWFsomething "QQSPBDI that &NFSZWJMMF is only ,FZ the $VSSJDVMVN privilege 1SFTTof some select by research mathematicians. In that sense, this mathematically endowed individuals. 4UJFHFS 4 4XBNJ 7 5JNF UP MFU HP /P BVUPNBUJD BFTUIFUJD QSFGFSFODF GPS UIF HPMEFO SBUJP JO BSU QJDUVSFT 1TZDIPMPHZ PG "FTUIFUJDT $SFBUJWJUZ BOE UIF "SUT 7PM 'FC IUUQEYEPJPSHB
8BMTFS ) DzF (PMEFO 4FDUJPO 8BTIJOHUPO %$ ɨF .BUIFNBUJDBM "TTPDJBUJPO PG "NFSJDB Reference:
1. Darvas, G. (2001). Symmetry. Basel: Birkhäuser Verlag. Figure 21: Cutting off two rhombi and a golden trapezium &OEOPUFT Figure 21. Cutting of two rhombi and a golden 2. De Villiers, M. (2009). Some Adventures in Euclidean Geometry. Lulu Press. ɨJT JT OPU UIF DPNNPO UFYUCPPL EFëOJUJPO ɨF VTVBM EFëOJUJPO JT " QBSBMMFMPHSBN JT B GPVSTJEFE ëHVSF GPS XIJDI CPUI QBJST PG trapezium 3. De Villiers, M. (2011). Equi-angled cyclic and equilateral circumscribed polygons. Te Mathematical Gazette, 95(532), By cutting off two rhombi and a golden isosceles trapezium as shown in Figure 21, we PQQPTJUFMarch, pp. TJEFT 102-106. BSF QBSBMMFM Accessed UP FBDI 16 October PUIFS *2016 XBOU at: UP http://dynamicmathematiclearning.com/equi-anglecyclicpoly.pdf FNQIBTJ[F UIBU DPODFQUT DBO CF EFëOFE EJêFSFOUMZ BOE PGUFO NPSF QPXFSGVMMZ JO Figure 20: A golden hexagon with adjacent sides in golden ratio UFSNT PG TZNNFUSZ "T BSHVFE JO %F 7JMMJFST JU JT NPSF DPOWFOJFOU EFëOJOH RVBESJMBUFSBMT JO UFSNT PG TZNNFUSZ UIBO UIF By cutting of two rhombi and a golden isosceles 4. De Villiers, M. (2016). Enrichment for the Gifted: Generalizing some Geometrical Teorems & Objects. Learning Figure 20. A golden hexagonalso withobtain adjacent a similar golden cyclic hexagon. Lastly, it is also left to the reader to consider, TUBOEBSE UFYUCPPL EFëOJUJPOT 3FGFSFODF %F 7JMMJFST . 4JNQMZ 4ZNNFUSJD .BUIFNBUJDT 5FBDIJOH .BZ Qo trapezium as shown in Figure 21, we also obtain a and Teaching Mathematics, December 2016. Accessed 16 October 2016 at: http://dynamicmathematicslearning.com/ Constructively Defining a ‘Golden Cyclic Hexagon’ sides in goldendefine ratio and investigate an analogoussimilar golden dual cyclic of a hexagon. golden cyclic Lastly, hexagon. it is also left ɨFICME13-TSG4-generalization.pdf (PMEFO 3BUJP DBO CF EFëOFE JO EJêFSFOU XBZT ɨF TJNQMFTU POF JT JU JT UIBU QPTJUJWF OVNCFS Y GPS XIJDI Y = + /Y √ Before closing, I’d like to briefly tease the reader with considering investigatingto the reader defining to consider, defne and investigate 5. FRVJWBMFOUMZ De Villiers, M. UIBU (2017). QPTJUJWF An OVNCFSexample YofGPS constructive XIJDI Y de=fYning:+ From ɨF EFëOJUJPOa golden rectangle JNQMJFT to UIBU goldenY =( quadrilaterals + )/ & XIPTF WBMVF JT BQQSPYJNBUFMZ " SFDUBOHMF XIPTF MFOHUI : XJEUI SBUJP JT Y : JT LOPXO BT B HPMEFO SFDUBOHMF *U IBT UIF GFBUVSF UIBU XIFO Constructively Defning a Concluding Remarks an analogous dual of a golden cyclic hexagon. beyond: Part 1. At Right Angles. Volume 6, No. 1, (March), pp. 64-69. hexagonal ‘golden’ analogues for at least some of the golden quadrilaterals discussed XF SFNPWF UIF MBSHFTU QPTTJCMF TRVBSF GSPN JU B CZ TRVBSF UIF SFDUBOHMF UIBU SFNBJOT JT BHBJO B HPMEFO SFDUBOHMF ‘Golden Cyclic Hexagon’ Though most of the mathematical results discussed here are not novel, it is hoped that this 6. Olive, J. (Undated). Construction and Investigation of Golden Trapezoids. University of Georgia, Athens; ɨFUSA: UFSN Classroom (PMEFO Notes. 3FDUBOHMF Accessed IBT CZ23 OPXOctober B TUBOEBSE 2016 at: NFBOJOH http://math.coe.uga.edu/olive/EMAT8990FYDS08/ )PXFWFS UFSNT MJLF (PMEFO 3IPNCVT (PMEFO 1BSBMMFMPHSBN (PMEFO here. For example, theBefore analogous closing, I’d equivalent like to brie off y atease rectangle the reader is an equi-angled, cyclic little investigation has toConcluding some extent Remarks shown the productive process of constructive 5SBQF[JVNGoldenTrapezoids.doc BOE (PMEFO ,JUF IBWF CFFO EFëOFE JO TMJHIUMZ EJêFSFOU XBZT CZ EJêFSFOU BVUIPST 6 hexagon as pointed with out considering in De Villiers defning (2011; hexagonal 2016). ‘golden’ Hence, one possibleTough most way of tothe mathematical results analogues for at least some ofdefining the golden by illustrating how new mathematical objects can be defined and constructed construct a hexagonal analogue for the golden rectangle is to impose thediscussed condition here on are an not novel, it is hoped that this quadrilaterals discussed here.from For example,familiar thedefinitions littleof known investigation objects. has In to the some pr ocess,extent shownseveral different possibilities analogous equivalent of a rectangle is an equi- MICHAEL DE VILLIERS has worked as researcher, mathematics and science teacher at institutions across equi-angled, cyclic hexagon that all1 the pairs of adjacent sides as shownthe in Figureproductive 19 areprocess of constructive defning MICHAEL DE VILLIERS 6 may be explored and compared in terms of the number of properties, ease of construction the world. Since 1991 he has workedbeen part as researcher,of the University mathematics of Durban-Westville and science teacher (now at institutions University across of KwaZulu- the angled, cyclic hexagon as pointed out in De world. He retired from the University of KwaZulu-Natal at the end of January 2016. He has subsequently been in the golden ratio; i.e., a ‘golden (cyclic) hexagon’. It is left to the readerby to illustrating verify that how if new mathematical objects Natal). He was editor of Pythagoras, the research journal of the Association of Mathematics Education of South Villiers (2011; 2016). Hence,or oneof proof,possible and way, in this particularcan be def case,ned and perh constructedaps also offrom visual familiar appeal. Moreover, it was appointed Professor Extraordinaire in Mathematics Education at the University of Stellenbosch. He was editor of to construct a7 hexagonal analogue for the golden Africa,Pythagoras and, the has research been vice-chairjournal of theof Associationthe SA Mathematics of Mathematics Olympiad Education since of 1997.South Africa,His main and wasresearch vice-chair interests of are FA/AB = phi, then AL/LM = phi , etc. In other words, the main diagonalsdefnitions divide of eachknown objects. In the process, Geometry, Proof, Applications and Modeling, Problem Solving, and the History of Mathematics. His home rectangle is to impose the conditionshown how on an some definitions of the same object might be more convenient than others in the SA Mathematics Olympiad from 1997 -2015, but is still involved. His main research interests are Geometry, other into the golden ratio. several diferent possibilities may be explored and pageProof, isApplications http://dynamicmathematics-learning.com/homepage4.html and Modeling, Problem Solving, and the History of. Mathematics. He maintains His a home web page page is forhttp:// dynamic terms of the deductive derivation of other properties not contained in the definition. geometrydynamicmathematics-learning.com/homepage4.html sketches at http://dynamic-mathematicslearning.com/JavaGSPLinks.htm. He maintains a web page for dynamic. He geometry may be sketches contacted on The observant reader would also note that ABEF, ABCD and CDEF, are all three [email protected] http://dynamic-mathematicslearning.com/JavaGSPLinks.htm. . He may be contacted on [email protected]. The process of constructive defining also generally applies to the definition and golden trapezia of the5 Olive type (undated), constructed for example, constructively and defined defnes two in di f theerent, firsinterestingt case types of in golden Figure trapezoids/trapeziums. 8. 6 Tis type of hexagon is also called a semi-regularexploration angle-hexagon of different in the referenced axiom papers. systems in pure, mathematical research where quite often Moreover, ALNF, ABCF7 It was, with etc. surprised, are goldeninterest that trapeziain October 2016, of theI came second upon Odom’s type construction constructed at: http://demonstrations.wolfram.com/ and HexagonsAndTeGoldenRatio/, whichexisting is the converse axiom of this result. systems With reference are used to the fgure, as startingOdom’s construction blocks involves which extending are the then sides modified, adapted, defined in Figure 8. of the equilateral triangle LMN to construct three equilateral triangles ABL, CDM and EFN. If the extension is proportional to the golden ratio, then the outer vertices of these three trianglesgeneralized, determine a etc.(cyclic,, toequi-angled) create hexagon and explore with adjacent new sides inmathematical the golden ratio. theories. So this little episode encapsulates at an elementary level some of the main research methodologies used by 80 At Right Angles | Vol. 6, No. 2, August 2017 At Right Angles | Vol. 6, No. 2, August 2017 81 research mathematicians. In that sense, this investigation has hopefully also contributed a 5 Olive (undated), for example, constructively defines two different, interesting types of golden trapezoids/trapeziums. little bit to demystifying where definitions come from, and that they don’t just pop out of 6 This type of hexagon is also called a semi-regular angle-hexagon in the referenced papers. 7 the air into a mathematician’s mind or suddenly magically appear in print in a school It was with surprised interest that in October 2016, I came upon Odom’s construction at: At Right Angles | Vol. 6, No. 2, August 2017 PB http://demonstrations.wolfram.com/HexagonsAndTheGoldenRatio/textbook., which is the converse of this result. With reference to the figure, Odom’s construction involves extending the sides of the equilateral triangle LMN to construct three equilateral triangles ABL, CDM and EFN. InIf the a extension classroom is proportional context, ifto the a teacher were to ask students to suggest various golden ratio, then the outer vertices of these three triangles determine a (cyclic, equi-angled) hexagon with At Right Angles | Vol. 6, No. 1, March 2017 69 adjacent sides in the golden ratio. possible definitions for golden quadrilaterals or golden hexagons of different types, it is Constructively Defning Other equi-angled, cyclic hexagon that all the pairs of compared in terms of the number of properties, investigation has hopefully also contributed a ‘Golden Quadrilaterals’ adjacent sides as shown in Figure 19 are in the ease of construction or of proof, and, in this little bit to demystifying where defnitions come Tis investigation has already become longer than golden ratio; i.e., a ‘golden (cyclic) hexagon’. It particular case in relation to the golden ratio, from, and that they don’t just pop out of the I’d initially anticipated, and it is time to fnish it is left to the reader to verify that if FA/AB = phi, perhaps also of visual appeal. Moreover, it was air into a mathematician’s mind or suddenly 7 of before I start boring the reader. Moreover, my then AL/LM = phi , etc. In other words, the main shown how some defnitions of the same object magically appear in print in a school textbook. main objective of showing constructive defning diagonals divide each other into the golden ratio. might be more convenient than others in terms of 3FGFSFODFTthe deductive derivation of other properties not In a classroom context, if a teacher were to ask in action has hopefully been achieved by now. Te observant reader would also note that ABEF, students to suggest various possible defnitions contained%F 7JMMJFST in .the defnition.4PNF "EWFOUVSFT JO &VDMJEFBO (FPNFUSZ -VMV 1SFTT However, I’d like to point out that there are ABCD and CDEF, are all three golden trapezia for golden quadrilaterals or golden hexagons several other types of quadrilaterals for which of the type constructed and defned in the frst Te&SOFTU process 1 of constructiveDzF 1IJMPTPQIZ def PGning .BUIFNBUJDT also generally &EVDBUJPO -POEPO of di 'BMNFSferent 1SFTT types, it is likely that they would case in Figure 8. Moreover, ALNF, ABCF, etc., applies to the defnition and exploration of propose several of the examples discussed here, one can similarly explore ways to defne ‘golden 'SFVEFOUIBM ) .BUIFNBUJDT BT BO &EVDBUJPOBM 5BTL % 3FJEFM %PSESFDIU )PMMBOE quadrilaterals’, e.g., cyclic quadrilaterals, are golden trapezia of the second type constructed diferent axiom systems in pure, mathematical and perhaps even a few not explored here. circumscribed quadrilaterals, trapeziums5, and defned in Figure 8. research(SPTTNBO where 1 FU quite BM often %P existing 1FPQMF 1SFGFSaxiom *SSBUJPOBM 3BUJPT "Involving /FX -PPL students BU UIF (PMEFO in an 4FDUJPO activity 4UVEFOU like SFTFBSDIthis would DPOEVDUFE JO JO UIF %FQU PG "QQMJFE $PNQVUFS 4DJFODF 6OJWFSTJUZ PG #BNCFSH "DDFTTFE PO 0DU BU 5 systems are used as starting blocks which are then not only more realistically simulate actual quadrilaterals, circumscribedbi-centric quadrilaterals, quadrilaterals, orthodiagonal trapeziums , bi-centric quadrilaterals, IUUQTXXXBDBEFNJBFEVɨF@(PMEFO@3BUJP quadrilaterals, equidiagonal quadrilaterals, etc. modifed, adapted, generalized, etc., to create mathematical research, but also provide students orthodiagonal quadrilaterals, equidiagonal quadrilaterals, etc. and-PFC explore "- new 7BSOFZ mathematical 8 %PFTtheories. UIF (PMEFO So this 4QJSBM &YJTU with BOE *G a /PU more 8IFSF personal JT JUT $FOUFS sense of *O )BSHJUUBJ ownership * over 1JDLPWFS the $" little episode4QJSBM encapsulates 4JNJMBSJUZ 4JOHBQPSF at an elementary 8PSME 4DJFOUJëD level QQ mathematical content instead of being seen as some4FSSB of .the main &EJUJPOresearch methodologies%JTDPWFSJOH (FPNFUSZ used "O *OWFTUJHBUJWFsomething "QQSPBDI that &NFSZWJMMF is only ,FZ the $VSSJDVMVN privilege 1SFTTof some select by research mathematicians. In that sense, this mathematically endowed individuals. 4UJFHFS 4 4XBNJ 7 5JNF UP MFU HP /P BVUPNBUJD BFTUIFUJD QSFGFSFODF GPS UIF HPMEFO SBUJP JO BSU QJDUVSFT 1TZDIPMPHZ PG "FTUIFUJDT $SFBUJWJUZ BOE UIF "SUT 7PM 'FC IUUQEYEPJPSHB
8BMTFS ) DzF (PMEFO 4FDUJPO 8BTIJOHUPO %$ ɨF .BUIFNBUJDBM "TTPDJBUJPO PG "NFSJDB Reference:
1. Darvas, G. (2001). Symmetry. Basel: Birkhäuser Verlag. Figure 21: Cutting off two rhombi and a golden trapezium &OEOPUFT Figure 21. Cutting of two rhombi and a golden 2. De Villiers, M. (2009). Some Adventures in Euclidean Geometry. Lulu Press. ɨJT JT OPU UIF DPNNPO UFYUCPPL EFëOJUJPO ɨF VTVBM EFëOJUJPO JT " QBSBMMFMPHSBN JT B GPVSTJEFE ëHVSF GPS XIJDI CPUI QBJST PG trapezium 3. De Villiers, M. (2011). Equi-angled cyclic and equilateral circumscribed polygons. Te Mathematical Gazette, 95(532), By cutting off two rhombi and a golden isosceles trapezium as shown in Figure 21, we PQQPTJUFMarch, pp. TJEFT 102-106. BSF QBSBMMFM Accessed UP FBDI 16 October PUIFS *2016 XBOU at: UP http://dynamicmathematiclearning.com/equi-anglecyclicpoly.pdf FNQIBTJ[F UIBU DPODFQUT DBO CF EFëOFE EJêFSFOUMZ BOE PGUFO NPSF QPXFSGVMMZ JO Figure 20: A golden hexagon with adjacent sides in golden ratio UFSNT PG TZNNFUSZ "T BSHVFE JO %F 7JMMJFST JU JT NPSF DPOWFOJFOU EFëOJOH RVBESJMBUFSBMT JO UFSNT PG TZNNFUSZ UIBO UIF By cutting of two rhombi and a golden isosceles 4. De Villiers, M. (2016). Enrichment for the Gifted: Generalizing some Geometrical Teorems & Objects. Learning Figure 20. A golden hexagonalso withobtain adjacent a similar golden cyclic hexagon. Lastly, it is also left to the reader to consider, TUBOEBSE UFYUCPPL EFëOJUJPOT 3FGFSFODF %F 7JMMJFST . 4JNQMZ 4ZNNFUSJD .BUIFNBUJDT 5FBDIJOH .BZ Qo trapezium as shown in Figure 21, we also obtain a and Teaching Mathematics, December 2016. Accessed 16 October 2016 at: http://dynamicmathematicslearning.com/ Constructively Defining a ‘Golden Cyclic Hexagon’ sides in goldendefine ratio and investigate an analogoussimilar golden dual cyclic of a hexagon. golden cyclic Lastly, hexagon. it is also left ɨFICME13-TSG4-generalization.pdf (PMEFO 3BUJP DBO CF EFëOFE JO EJêFSFOU XBZT ɨF TJNQMFTU POF JT JU JT UIBU QPTJUJWF OVNCFS Y GPS XIJDI Y = + /Y √ Before closing, I’d like to briefly tease the reader with considering investigatingto the reader defining to consider, defne and investigate 5. FRVJWBMFOUMZ De Villiers, M. UIBU (2017). QPTJUJWF An OVNCFSexample YofGPS constructive XIJDI Y de=fYning:+ From ɨF EFëOJUJPOa golden rectangle JNQMJFT to UIBU goldenY =( quadrilaterals + )/ & XIPTF WBMVF JT BQQSPYJNBUFMZ " SFDUBOHMF XIPTF MFOHUI : XJEUI SBUJP JT Y : JT LOPXO BT B HPMEFO SFDUBOHMF *U IBT UIF GFBUVSF UIBU XIFO Constructively Defning a Concluding Remarks an analogous dual of a golden cyclic hexagon. beyond: Part 1. At Right Angles. Volume 6, No. 1, (March), pp. 64-69. hexagonal ‘golden’ analogues for at least some of the golden quadrilaterals discussed XF SFNPWF UIF MBSHFTU QPTTJCMF TRVBSF GSPN JU B CZ TRVBSF UIF SFDUBOHMF UIBU SFNBJOT JT BHBJO B HPMEFO SFDUBOHMF ‘Golden Cyclic Hexagon’ Though most of the mathematical results discussed here are not novel, it is hoped that this 6. Olive, J. (Undated). Construction and Investigation of Golden Trapezoids. University of Georgia, Athens; ɨFUSA: UFSN Classroom (PMEFO Notes. 3FDUBOHMF Accessed IBT CZ23 OPXOctober B TUBOEBSE 2016 at: NFBOJOH http://math.coe.uga.edu/olive/EMAT8990FYDS08/ )PXFWFS UFSNT MJLF (PMEFO 3IPNCVT (PMEFO 1BSBMMFMPHSBN (PMEFO here. For example, theBefore analogous closing, I’d equivalent like to brie off y atease rectangle the reader is an equi-angled, cyclic little investigation has toConcluding some extent Remarks shown the productive process of constructive 5SBQF[JVNGoldenTrapezoids.doc BOE (PMEFO ,JUF IBWF CFFO EFëOFE JO TMJHIUMZ EJêFSFOU XBZT CZ EJêFSFOU BVUIPST 6 hexagon as pointed with out considering in De Villiers defning (2011; hexagonal 2016). ‘golden’ Hence, one possibleTough most way of tothe mathematical results analogues for at least some ofdefining the golden by illustrating how new mathematical objects can be defined and constructed construct a hexagonal analogue for the golden rectangle is to impose thediscussed condition here on are an not novel, it is hoped that this quadrilaterals discussed here.from For example,familiar thedefinitions littleof known investigation objects. has In to the some pr ocess,extent shownseveral different possibilities analogous equivalent of a rectangle is an equi- MICHAEL DE VILLIERS has worked as researcher, mathematics and science teacher at institutions across equi-angled, cyclic hexagon that all1 the pairs of adjacent sides as shownthe in Figureproductive 19 areprocess of constructive defning MICHAEL DE VILLIERS 6 may be explored and compared in terms of the number of properties, ease of construction the world. Since 1991 he has workedbeen part as researcher,of the University mathematics of Durban-Westville and science teacher (now at institutions University across of KwaZulu- the angled, cyclic hexagon as pointed out in De world. He retired from the University of KwaZulu-Natal at the end of January 2016. He has subsequently been in the golden ratio; i.e., a ‘golden (cyclic) hexagon’. It is left to the readerby to illustrating verify that how if new mathematical objects Natal). He was editor of Pythagoras, the research journal of the Association of Mathematics Education of South Villiers (2011; 2016). Hence,or oneof proof,possible and way, in this particularcan be def case,ned and perh constructedaps also offrom visual familiar appeal. Moreover, it was appointed Professor Extraordinaire in Mathematics Education at the University of Stellenbosch. He was editor of to construct a7 hexagonal analogue for the golden Africa,Pythagoras and, the has research been vice-chairjournal of theof Associationthe SA Mathematics of Mathematics Olympiad Education since of 1997.South Africa,His main and wasresearch vice-chair interests of are FA/AB = phi, then AL/LM = phi , etc. In other words, the main diagonalsdefnitions divide of eachknown objects. In the process, Geometry, Proof, Applications and Modeling, Problem Solving, and the History of Mathematics. His home rectangle is to impose the conditionshown how on an some definitions of the same object might be more convenient than others in the SA Mathematics Olympiad from 1997 -2015, but is still involved. His main research interests are Geometry, other into the golden ratio. several diferent possibilities may be explored and pageProof, isApplications http://dynamicmathematics-learning.com/homepage4.html and Modeling, Problem Solving, and the History of. Mathematics. He maintains His a home web page page is forhttp:// dynamic terms of the deductive derivation of other properties not contained in the definition. geometrydynamicmathematics-learning.com/homepage4.html sketches at http://dynamic-mathematicslearning.com/JavaGSPLinks.htm. He maintains a web page for dynamic. He geometry may be sketches contacted on The observant reader would also note that ABEF, ABCD and CDEF, are all three [email protected] http://dynamic-mathematicslearning.com/JavaGSPLinks.htm. . He may be contacted on [email protected]. The process of constructive defining also generally applies to the definition and golden trapezia of the5 Olive type (undated), constructed for example, constructively and defined defnes two in di f theerent, firsinterestingt case types of in golden Figure trapezoids/trapeziums. 8. 6 Tis type of hexagon is also called a semi-regularexploration angle-hexagon of different in the referenced axiom papers. systems in pure, mathematical research where quite often Moreover, ALNF, ABCF7 It was, with etc. surprised, are goldeninterest that trapeziain October 2016, of theI came second upon Odom’s type construction constructed at: http://demonstrations.wolfram.com/ and HexagonsAndTeGoldenRatio/, whichexisting is the converse axiom of this result. systems With reference are used to the fgure, as startingOdom’s construction blocks involves which extending are the then sides modified, adapted, defined in Figure 8. of the equilateral triangle LMN to construct three equilateral triangles ABL, CDM and EFN. If the extension is proportional to the golden ratio, then the outer vertices of these three trianglesgeneralized, determine a etc.(cyclic,, toequi-angled) create hexagon and explore with adjacent new sides inmathematical the golden ratio. theories. So this little episode encapsulates at an elementary level some of the main research methodologies used by 80 At Right Angles | Vol. 6, No. 2, August 2017 At Right Angles | Vol. 6, No. 2, August 2017 81 research mathematicians. In that sense, this investigation has hopefully also contributed a 5 Olive (undated), for example, constructively defines two different, interesting types of golden trapezoids/trapeziums. little bit to demystifying where definitions come from, and that they don’t just pop out of 6 This type of hexagon is also called a semi-regular angle-hexagon in the referenced papers. 7 the air into a mathematician’s mind or suddenly magically appear in print in a school It was with surprised interest that in October 2016, I came upon Odom’s construction at: At Right Angles | Vol. 6, No. 2, August 2017 PB http://demonstrations.wolfram.com/HexagonsAndTheGoldenRatio/textbook., which is the converse of this result. With reference to the figure, Odom’s construction involves extending the sides of the equilateral triangle LMN to construct three equilateral triangles ABL, CDM and EFN. InIf the a extension classroom is proportional context, ifto the a teacher were to ask students to suggest various golden ratio, then the outer vertices of these three triangles determine a (cyclic, equi-angled) hexagon with At Right Angles | Vol. 6, No. 1, March 2017 69 adjacent sides in the golden ratio. possible definitions for golden quadrilaterals or golden hexagons of different types, it is