Theory of Knowledge - Course Companion for Students Marija Uzunova Dang Arvin Singh Uzunov Dang
gomtry was dismissd forHyproliof yars as impossil, aus ithundrdsEulid’s axiom aout paralll lins. Itviolatsalso xtrmly diffiult to visualiz orwasvn using omputrs. Daina Taiminamodl,using roht to onstrut tangil 3Dganthat mad it asir to omprhnd. If youmodlsurious aout tatil mathmatis andarroht, you an follow th link tohyproliout mor.findtrms: CrohtingSarhplans YouTuhyproliroht modls, ragas and sa slugs inWithw should ask to what xtnt mathmatismind, modid knowldg. Wrthimanit as a knowing that mrgs fromdsrisprforming mathmatis, a kind ofhandsfiguring—ut would shools andmodidhsitat to us roht in advandtahrslasss? Do algrai quationsmathmatismor or lss valid as a tahing tool? Isflfling drivd from a partiular ulturalthatshap shown in Figur 11.8 onsists of 4Thdodagons and 12 dagons, with 28rgulargaps in th shap of quilatral triangls.littlit is also an impossil shap, withHowvr,that will not mt at th dgs. Itpolygonsnot al to los, ut it works ausshouldatgory of almost prft mathmatisThisalld nar-miss mathmatis. Thr is noisdfinition of a nar miss, ut Craigprislif is that th mathmatialKaplan’sis omparal to pratial rrorsrrorfrom using “ral-world matrials andarisingimprft hands” (Kaplan quotd inyourLam2017).xampl of nar-miss mathmatisAnothrth missing-squar puzzl (s Figur 11.9).isA is ut into four pis and arrangdTriangltriangl B, ut suddnly a gap appars.intoan this ? It is anothr nar miss: thHowar not atually triangls, as thtrianglsis not a straight lin—th gradinthypotnusfrom 0.4 (lu hypotnus) to 0.375hangshypotnus). To what xtnt do nar(rddiminish or add to our knowldg aoutmisssIs thr valu in pratial, utmathmatis?mathmatis that annot gaindimpossil,III. Methods and toolsIII. Methods and toolsFigure 11.8 A mathmatially impossil shap, mad possil onlyaus of imprptil warping of th paprontxt?III.5 Doing impossible mathematicsfrom astrat mathmatis?of vry slight warping of th papr.AThe fudge factor that arises just from working in thereal world with paper means that things that ought tobe impossible actually aren’t.(Kaplan quoted in Lamb 2017)BFigure 11.9 Th missing-squar puzzl345
III. Methods and toolswritrs of The Simpsons tlvisionThsomtims ommuniat with thirsrisviwrs y asuallymathmatially-inlindnar misss into th akgrounddroppingmathmatis is mor than a uriosity orNar-missfor pratial joks. Th rason pianos havasiskys in an otav, for xampl, is du to a nar12Th two most important musial intrvalsmiss.an otav (a frquny ratio of 2:1) and a fifth (aarof 3:2), ut it is impossil to divid an otavratioa way that nsurs all th fifths will prft.inis mathmatially impossil to ronil thItfrqunis (tons) of otavs and fifths.diffrntyou an gt vry los y dividing th otav“But12 qual half-stps, svn of whih giv youintofrquny ratio of 1.498. That’s good nough foraalso th Ramanujan onstant: e π √163 ,Considralmost quals a whol numr:whihsns. In th pisod “Th Wizard ofofTrra”, Homr Simpson writsEvrgrnth quation on thir alulators mayhkdn shokd to find that Homr Simpsonhavdisprovd Frmat’s last thorm! In fat,hadalulators ar not pris nough tomostis a vry nar-miss. Frmat’s lastwhihis saf.thormwritrs of The Simpsons ar known forThrfrns to advand mathmatisinludingth show. Simon Singh, author of Fermat’sin(1998) and The Simpsons and TheirEnigmaSecrets (2013), xplains furthr inMathematicallinkd vido.thtrms: Homr’s lastSarhYouTuthormHow262,537,412,640,768,743.99999999999925.thr irrational numrs omin to form adonumr? Could it a lu to a dprrationalof mathmatis? Th mathmatiian Johnpimisss hav inspird pratial appliationsNarwll as uriosity in mathmatiians to lookasand dig dpr. Thy an srv as luslosrwhat disovris might tru, yaoutalmost tru. Mathmatiians disovringmisss through xprimntation, play,narimprftions, and trial and rror—ral-worldthat ar not typially assoiatd withwords11Box 11.4: D’oh! That time Homer Simpson nearly solved Fermat’s last theoremout th quation3987 12 + 4365 12 = 4472 12violats th thorm that a n + b n = c nwhihno intgr solution if n > 2. Viwrs whohasshow th lft sid of th quation:3,987 12 + 4,365 12 = 4,472.0000000070576171875 12Baz, among othrs, thinks so.most popl” (Lam 2017).mathmatialmthods.346
- Page 301 and 302: II. Perspectivesterms: CNN Business
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gomtry was dismissd for
Hyproli
of yars as impossil, aus it
hundrds
Eulid’s axiom aout paralll lins. It
violats
also xtrmly diffiult to visualiz or
was
vn using omputrs. Daina Taimina
modl,
using roht to onstrut tangil 3D
gan
that mad it asir to omprhnd. If you
modls
urious aout tatil mathmatis and
ar
roht, you an follow th link to
hyproli
out mor.
find
trms: Crohting
Sarh
plans YouTu
hyproli
roht modls, ragas and sa slugs in
With
w should ask to what xtnt mathmatis
mind,
modid knowldg. Wrthim
an
it as a knowing that mrgs from
dsris
prforming mathmatis, a kind of
hands
figuring—ut would shools and
modid
hsitat to us roht in advand
tahrs
lasss? Do algrai quations
mathmatis
mor or lss valid as a tahing tool? Is
fl
fling drivd from a partiular ultural
that
shap shown in Figur 11.8 onsists of 4
Th
dodagons and 12 dagons, with 28
rgular
gaps in th shap of quilatral triangls.
littl
it is also an impossil shap, with
Howvr,
that will not mt at th dgs. It
polygons
not al to los, ut it works aus
should
atgory of almost prft mathmatis
This
alld nar-miss mathmatis. Thr is no
is
dfinition of a nar miss, ut Craig
pris
lif is that th mathmatial
Kaplan’s
is omparal to pratial rrors
rror
from using “ral-world matrials and
arising
imprft hands” (Kaplan quotd in
your
Lam2017).
xampl of nar-miss mathmatis
Anothr
th missing-squar puzzl (s Figur 11.9).
is
A is ut into four pis and arrangd
Triangl
triangl B, ut suddnly a gap appars.
into
an this ? It is anothr nar miss: th
How
ar not atually triangls, as th
triangls
is not a straight lin—th gradint
hypotnus
from 0.4 (lu hypotnus) to 0.375
hangs
hypotnus). To what xtnt do nar
(rd
diminish or add to our knowldg aout
misss
Is thr valu in pratial, ut
mathmatis?
mathmatis that annot gaind
impossil,
III. Methods and tools
III. Methods and tools
Figure 11.8 A mathmatially impossil shap, mad possil only
aus of imprptil warping of th papr
ontxt?
III.5 Doing impossible mathematics
from astrat mathmatis?
of vry slight warping of th papr.
A
The fudge factor that arises just from working in the
real world with paper means that things that ought to
be impossible actually aren’t.
(Kaplan quoted in Lamb 2017)
B
Figure 11.9 Th missing-squar puzzl
345
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