Art Meets Mathematics in the Fourth Dimension

191 Pages·2014·7.22 MB·English

by  Stephen Lipscomb

Stephen Leon Lipscomb
Art Meets
Mathematics in
the Fourth
Dimension
Second Edition
Art Meets Mathematics in the Fourth Dimension

Stephen Leon Lipscomb
Art Meets Mathematics
in the Fourth Dimension
Second Edition
123
StephenLeonLipscomb
EmeritusProfessorofMathematics
UniversityofMaryWashington
Fredericksburg,VA,USA
GeraldEdgarsuggestedtheArtMeetsMathematicsintheFourthDimensiontitle.
Videostothisbookcanbeaccessedat
http://www.springerimages.com/videos/978-3-319-06253-2
ISBN978-3-319-06253-2 ISBN978-3-319-06254-9(eBook)
DOI10.1007/978-3-319-06254-9
SpringerChamHeidelbergNewYorkDordrechtLondon
LibraryofCongressControlNumber:2014940943
MathematicsSubjectClassification:28A80,55M10,97M80
SpringerInternationalPublishingSwitzerland2011, 2014
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Dedicated to
James Cornelius Perry
August 9, 1935 – January 10, 2011
Circa2000, Perrycreatedan isotopy thatmoves the4-web fromthe4th
dimension into humanview whilepreservingits fractal dimension
andexposing its self-similarity.
and
Eugene LeRoy Miller
November 5, 1939 – September 4, 2012
August2000, Miller creates firstphysical-art representationof the4-web.
Basic4-webStructurefromthe
Fourth-Dimension (Graphic by Chris Dupilka)
“The most beautiful thing we can experience is the mysterious.
It is the source of all true art and science.”
Albert Einstein
AsaninstanceofEinstein’sstatement,considerthemysteriousquestion.
Howcanweusearttopicturetheintersectionofalineanda3-sphere?
Knowingthatthe3-sphereisa3-manifold,onemayincorrectlyguess
Foralocalview,pictureaneedleinanicecube.
But(section)40showsthatalineintersectsa3-sphereinatmosttwopoints. So the
correctansweris
Threeworksofart(pictures)arerequired: onecontainingnothing,another
containingonedotofink,andanotherthatcontainstwodotsofink.
The point is, pictures can convey “mysterious mathematical facts” to even
non-mathematicians,andthistruismisthekeystoneofthisbook.
Preface (Second Edition)
Thetitleof the2011 firstedition is “Incipit” (herebegins), andits subtitleis
thetitleofthis2014secondedition. Otheradjustmentsincludethefootnote
onpage68where“skull”replaces“scull”;thetopofpage79where“−t(2m·
b)”replaces“+t(2m·b)”;thethirdparagraphonpage80;themiddleofpage
82wherethePov-raycode-definitionof"sphere"iscorrected;andfinally,the
top-rightof page 118 where“(12/d)2” replaces “(12/d)”.
The first edition contains a Blu-ray HD Disc, and MPG-4 versions of the
videos therein are available at Springer web sites: The author’s narration
explainsthecoloringoftheGod’sImage? art,withbackgroundmusictheart
is rotated, and a 4-web grid from the 4th dimension is moved into human
view. Frames ofthesevideos areillustratedwithin thecolor-platesection.
Forthecreationofthissecondedition,theauthorgivesthankstoSpringer
editors AnnKostantandMarcStraussfortheirguidance and support.
Spotsylvania,VA Stephen L. Lipscomb
April 2014
About the Author
AgraduateofFairmontStateUniversityBA;WestVirginiaUniversityMA,andtheUni-
versity of Virginia Ph.D., where the late G. T. Whyburn (past president of the AMS) was
his advisor. And prior to Chairing the Department of Mathematics, Physics, and Com-
puter Science at the University of Mary Washington, Lipscomb was a U. S. Navy senior
mathematicianinvolved withtheTridentMissilesystemandserved as Team Chairof an
OperationalEvaluationoftheTomahawkMissilesystem.
In the 1970s the author solved a half-century old embedding problem in dimension
theory. Hissolution,uponreviewbyaTransactionsoftheAmericanMathematicalSociety
(AMS) referee, was called “an outstanding contribution to dimension theory”. In 2009
theauthor’sfeaturearticleTheQuestforUniversalSpacesinDimensionTheoryappeared
in the December issue of the Notices of the AMS, and his book Fractals and Universal
Spaces in Dimension Theory was published in the outstanding Springer Monographs in
MathematicsSeries. ItturnedoutthattheclassicalfractalsknownasSierpin´ski’striangle
and cheese were corollary to the author’s solution. And at the beginning of the 21st
Century,JamesPerryintroducedanotherexample(the4-web)oftheauthor’sconstruction
thatlivedintheFourthDimension. Perry’sexampleisfundamentaltothisbook.
During the 1990s, the author introduced an extension of Cauchy’s cycle notation in
grouptheorytopathnotationinsemi-grouptheory. Heusedhispathnotationtoconstruct
new semi-groups, e.g., the alternating semi-groups. By 1996, the author’s work with his
path notation was documented by theAMSin Volume 46 Symmetric Inverse Semigroups
ofitsprestigiousMathematicalSurveysandMonographsSeries.
Circa 2000 the author realized, because the triangle is theHoly Grail of strength de-
sign, that the 4-web would be ideal for practical applications. By 2005 he obtained a US
patentanddesignedmedical4-webspinecages. ThenJesseeHuntcreated4-Web,Inc.,the
cageseventuallybecameFDAapproved forhumanimplantation,andtheimplantsbegan
in2011. In2014theUSPatentOfficeapprovedhis5-WebPatent.
vii

Preface
Over time the impossible becomes understandable and partially possible.
Here begins new art created by capturing pictures of objects that live in the
4th dimension. The new method is applied to the hyper-sphere, which is
a higher-dimensional offspring of an ordinary sphere. We see an ordinary
sphere, also called a 2-sphere, each time we look at a full moon or the sur-
faceofabasketball,butthehyper-spherelives in the4th-dimensionbeyond
humanview. A hyper-sphereis also called a3-sphere.
The “global structure” of a hyper-sphere requires four dimensions, but
humanvisionisonly3-dimensional. Inotherwords,itishumanlyimpossible
to “see”ahyper-spherein the4th dimension.
Nevertheless, the historical value of the hyper-sphere cannot be over-
stated: Einstein used it in 1917 to describe the shape of our universe, and
Dante, circa 1300 AD, used it to describe the Christian afterlife. These de-
scriptionsarepresentedinchapters2,3,and4usingnumerousillustrations.
Over time, our understanding of the meaning of dimension evolved. By
1970 a 4-web (think of a spider’s web) from the 4th dimension emerged. In
the4-webcase,however,by2003itwasdiscoveredthatits“globalstructure”
could be moved into human view (see left-side graphic above and reference
[27]). The 4-web is an example of a construction that was applied in the
article “On Imbedding Finite-Dimensional Metric Spaces”, which I published
in the Transactions of the American Mathematical Society Vol. 211, 1975.
While the ‘human view” of the 4-web is a relatively recent discovery, the
idea of“reasoningwellwith images of objects”datesback tothelate1800s.
For example, within the introduction of Poincaré’s famous Analysis Situs
(1895), wefind
... geometry is the art of reasoning well with badly made figures. Yes,
without doubt, but with one condition. The proportions of the figures
might be grossly altered but their elements must not be interchanged
andmustconservetheirrelativesituation.
By moving the 4-web into human view, we “see” its structure: The line
segmentsthatcomprisethe4-webareorganizedintofivegroups,eachgroup
“just touching” the other four. As Poincaré might say, the five groups were
notinterchangedandtheirrelativesituation was conserved.
ix