Table Of ContentAngelo Alessandro Mazzotti
All Sides
to an Oval
Properties, Parameters,
and Borromini’s
Mysterious Construction
All Sides to an Oval
Angelo Alessandro Mazzotti
All Sides to an Oval
Properties, Parameters, and Borromini’s
Mysterious Construction
AngeloAlessandroMazzotti
IstitutodiIstruzioneSuperiore
“I.T.C.DiVittorio–I.T.I.Lattanzio”
Roma,Italy
ISBN978-3-319-39374-2 ISBN978-3-319-39375-9 (eBook)
DOI10.1007/978-3-319-39375-9
LibraryofCongressControlNumber:2017933470
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To Sophie, Lilian and Anton, my greatest joy
Acknowledgements
I should like to thank Lydia Colin for her proofreading, Edoardo Dotto for his
enthusiasm, Kim Williams for her encouragement and Sophie Püschmann for her
technicalandspiritualsupport.
vii
Contents
1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2 PropertiesofaPolycentricOval. . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.1 Four-CentreOvals. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
3 Ruler/CompassConstructionsofSimpleOvals. . . . . . . . . . . . . . . . 19
3.1 OvalswithGivenSymmetryAxisLines. . . . . . . . . . . . . . . . . . 20
3.2 OvalswithUnknownAxisLines. . . . . . . . . . . . . . . . . . . . . . . . 39
3.3 InscribingandCircumscribingOvals:TheFrameProblem. . . . . 49
3.4 TheStadiumProblemandtheRunningTrack. . . . . . . . . . . . . . 56
References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
4 ParameterFormulasforSimpleOvalsandApplications. . . . . . . . 61
4.1 ParameterFormulasforSimpleOvals. . . . . . . . . . . . . . . . . . . . 61
4.2 LimitationsfortheFrameProblem. . . . . . . . . . . . . . . . . . . . . . 80
4.3 MeasuringaFour-CentreOval. . . . . . . . . . . . . . . . . . . . . . . . . 85
4.4 ConcentricOvals. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
5 OptimisationProblemsforOvals. . . . . . . . . . . . . . . . . . . . . . . . . . 93
References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
6 RemarkableFour-CentreOvalShapes. . . . . . . . . . . . . . . . . . . . . . 101
References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
7 Borromini’sOvalsintheDomeofSanCarloalleQuattroFontane
inRome. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
7.1 The1998Survey. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
7.2 TheProjectfortheDome. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
7.2.1 TheDimensionsoftheDome.. . . . . . . . . . . .. . . . . . . . 124
7.2.2 TheImpostOvalandtheLanternOval. . . . . . . . . . . . . . 127
ix
x Contents
7.2.3 DeterminingtheHeightoftheRingsofCoffers. . . . . . . 131
7.2.4 TheOvalsoftheRingsofCoffers. . . . . . . . . . . . . . . . . 136
7.3 Conclusions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144
8 Ovalswith4nCentresandtheGroundPlanoftheColosseum. . . . 145
8.1 TheConstructionofOvalswith4nCentres. . . . . . . . . . . . . . . . 145
8.2 TheOvalsintheGroundPlanoftheColosseum. . . . . . . . . . . . 148
8.2.1 ReferencesandDataUsed:TheTwo-StepMethod. . . . . 148
8.2.2 TheFour-CentreOvalGuideline. . . . . . . . . . . . . . . . . . 149
8.2.3 FromaFour-CentretoanEight-CentreOval. . . . . . . . . 152
References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158
Appendix. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159
1
Introduction
Whenwritingaboutovalsthefirstthingtodoistomakesurethatthereaderknows
what you are talking about. The word oval has, both in common and in technical
language,anambiguousmeaning.Itmaybeanyshaperesemblingacirclestretched
fromtwooppositesides,sometimesevenmoretoonesidethantotheother.Whenit
comes to mathematics you have to be precise if you don’t want to talk about
ellipses,oraboutnon-convexshapes,oraboutformswithasinglesymmetryaxis.
Polycentric ovals are convex, with two symmetry axes, and are made of arcs of
circle connected in a way that allows for a common tangent at every connection
point.Thisformdoesn’thaveanelegantequationasdotheellipse,Cassini’sOval,
orCartesianOvals.Butithasbeenusedprobablymorethananyothersimilarshape
tobuildarches,bridges,amphitheatres,churchesandwindowswheneverthecircle
was considered not convenient or simply uninteresting. The ellipse is nature, it is
howtheplanetsmove,whiletheovalishuman,itisimperfect.Ithasoftenbeenan
artist’sattempttoapproximatetheellipse,tocomeclosetoperfection.Buttheoval
allowsforfreedom,becausechoicesofpropertiesandshapestoinscribeorcircum-
scribe can be made by the creator. The fusion between the predictiveness of the
circle and the arbitrariness of how and when this changes into another circle is
described in the biography of the violin-maker Martin Schleske: “Ovals describe
neither a mathematical function (as the ellipse does) nor an arbitrary shape. [...]
Two elements mesh here in a fantastic dialectic: familiarity and surprise. They
formaharmoniccontrast.[...]Inthisshapetheonecannotexistwithouttheother.”
(ourtranslationfromtheGerman,[15],pp.47–48).
Polycentric ovals are and have been used by architects, painters, craftsmen,
engineers,graphicdesignersandmanyotherartistsandspecialists,buttheknowl-
edgeneededtodrawthedesiredshapehasbeen—mostlyinthepast—eitherspread
bywordofmouthorfoundoutbymethodsoftrialanderror.Thequestionofwhat
wasknownaboutovalsinancienttimeshasregainedinterestinthelast20years.In
[4] the idea of a missing chapter about amphitheatres in Vitruvius treatise De
Architectura Libri Decem is put forward by Duvernoy and Rosin, while in [7]
Lo´pezMozoarguesthatatthetimetheEscorialwasbuilttheknow-howhadtobe
#SpringerInternationalPublishingAG2017 1
A.A.Mazzotti,AllSidestoanOval,DOI10.1007/978-3-319-39375-9_1
2 1 Introduction
better than whatappears lookingat the sourcesavailable today. In[6]the authors
showhowtheovalshapehasbeenusedinthedesignofSpanishmilitarydefense,
while in [8] the author suggests that methods of drawing ovals with any given
proportion were within reach, if not known, at the time Francesco Borromini
planned the dome for S.Carlo alle Quattro Fontane, since they only required a
basic knowledge of Euclidean geometry (the whole of Chap. 7 of this book is
dedicatedtothisconstruction).Inanycase,tracesofpolycentriccurvesdateback
thousands of years (see for example Huerta’s extensive work on oval domes [5]).
For everything that has come down to us in terms of treatises on the generic oval
shape(laformaovataasshecallsit),withconstructionsexplainedbythearchitects
intheirownwords,thebookbyZerlenga[17]isamust.
Theideaofthisbookisnottodisputewhetherandwhenpolycentricovalswere
usedinthepast.Itistocreateacompactandstructuredsetofdata,bothgeometric
andalgebraic,coveringthesingletopicofpolycentricovalsinallitsmathematical
aspects,andthentoillustratetwoveryimportantcasestudies.Basicconstructions
andequationshavebeenusedand/orderivedbythosewhoneededthem,butsucha
collectionhas—tothebestofourknowledge—neverbeenputtogether.Andthisis
why this book can help those using the oval shape to make objects and to design
buildings,aswellasthoseusingitasameansoftheirartisticcreation,tomasterand
optimise the shape to fit their technical and/or artistic requirements. The author
wentdeepintothesubjectandthebookcontainsmanyofhiscontributions,someof
whichapparentlynotinvestigatedbefore.
The style chosen is that of mathematical rigour combined with easy-to-follow
passages, and this could only have been done because basic Euclidean geometry,
analyticgeometry,trigonometryandcalculushavebeenused.Non-mathematicians
whocanbenefitfromtheconstructionsand/orformulasondisplaywillthusbeable,
with a bit of work, to understand where these constructions and formulas
comefrom.
Themainpublishedcontributionstothisworkhavebeen:theauthor’spaperon
theconstructionofovals[8],Rosin’spapersonfamousovalconstructions[11,13],
and onthe comparison ofan ellipse with an oval [11–14],Dotto’ssurvey on oval
constructions[2]andhisbookonHarmonicOvals[3],Lo´pezMozo’spaperonthe
variousgeneralpurposeconstructionsofpolycentricovalsinhistory[7],Ragazzo’s
works on ovals and polycentric curves [9, 10]—which triggered the author’s
interestinboththesubjects—andfinallythemonographontheColosseum[1].
TotallynewtopicsdisplayedinthismonographareConstructions10and11,the
organisedcollectionofformulasinChap.4,theinscriptionandcircumscriptionof
rectangles and rhombi with ovals, the Frame Problems and the constructions of
ovalsminimisingtheratioandthedifferenceoftheradiiforanychoiceofaxes.An
organizedapproachtotheproblemofnestedovalsisalsopresented,whattheauthor
calls the Stadium Problem, as well as a new oval form by the author. The main
contribution is though the study on Borromini’s dome in the church of San Carlo
alle Quattro Fontane in Rome: together with Margherita Caputo a whole new
hypothesis on the steps which Borromini took in his mysterious project for his
complicatedovaldomeisputforward.
