Logical and Geometrical Distance in Polyhedral Aristotelian Diagrams in Knowledge Representation

22 Pages·2017·1.2 MB·English

SS
symmetry
Article
Logical and Geometrical Distance in Polyhedral
Aristotelian Diagrams in Knowledge Representation
LorenzDemey1,*andHansSmessaert2
1 CenterforLogicandAnalyticPhilosophy,KULeuven,3000Leuven,Belgium
2 ResearchGrouponFormalandComputationalLinguistics,KULeuven,3000Leuven,Belgium;
[email protected]
* Correspondence:[email protected];Tel.:[phone]
AcademicEditor:NeilY.Yen
Received:28August2017;Accepted:27September2017;Published:29September2017
Abstract: Aristotelian diagrams visualize the logical relations among a finite set of objects.
These diagrams originated in philosophy, but recently, they have also been used extensively in
artificial intelligence, in order to study (connections between) various knowledge representation
formalisms. Inthispaper,wedeveloptheideathatAristoteliandiagramscanbefruitfullystudied
as geometrical entities. In particular, we focus on four polyhedral Aristotelian diagrams for the
B
Booleanalgebra ,viz.therhombicdodecahedron,thetetrakishexahedron,thetetraicosahedron
4
and the nested tetrahedron. After an in-depth investigation of the geometrical properties and
interrelationshipsofthesepolyhedraldiagrams,weanalyzethecorrelation(orlackthereof)between
logical(Hamming)andgeometrical(Euclidean)distanceineachofthesediagrams. Theoutcome
of this analysis is that the Aristotelian rhombic dodecahedron and tetrakis hexahedron exhibit
thestrongestdegreeofcorrelationbetweenlogicalandgeometricaldistance;thetetraicosahedron
performsworse;andthenestedtetrahedronhasthelowestdegreeofcorrelation. Finally,theseresults
areusedtoshednewlightontherelativestrengthsandweaknessesofthesepolyhedralAristotelian
diagrams,byappealingtothecongruenceprinciplefromcognitiveresearchondiagramdesign.
Keywords: logical geometry; Boolean algebra; knowledge representation; bitstrings; rhombic
dodecahedron; tetrakis hexahedron; tetraicosahedron; nested tetrahedron; Hamming distance;
Euclideandistance;congruenceprinciple
1. Introduction
Aristotelian diagrams visualize the logical relations among a finite set of elements from some
logical,lexicalorconceptualsystem. Thehistoricaloriginsofthesediagramscanultimatelybetraced
backtothelogicalworksofAristotleandhiscommentators[1]. Theoldestandmostwidely-known
exampleistheso-called‘squareofopposition’forthecategoricalstatementsfromAristotle’slogical
system of syllogistics; cf. Figure 1. Throughout history, distinguished philosophers such as John
Buridan,GottfriedLeibnizandGottlobFregehavemadeuseofsquares,butalsolarger,morecomplex
Aristoteliandiagramstoillustrateandexplaintheirtheorizing[2–4]. However,becauseoftheubiquity
ofthelogicalrelationsthattheyvisualize,Aristoteliandiagramsarenowalsousedinotherscientificand
engineeringdisciplines,suchascognitivescience[5,6],neuroscience[7],naturallanguageprocessing[8],
law[9–11],linguistics[12–15]andartificialintelligence. Withinthelatterfield,Aristoteliandiagrams
havebeenusedtostudyvariouslogic-basedapproachestoknowledgerepresentation,includingfuzzy
logic[16–20],modal-epistemiclogic[21–25]andprobabilisticlogic[26–28]. Furthermore,Aristotelian
diagrams are also used extensively to study (the connections between) other types of knowledge
representation formalisms, such as formal argumentation theory [29–32], fuzzy set theory [33–36],
formalconceptanalysisandpossibilitytheory[37–39],roughsettheory[37,40,41],multiple-criteria
Symmetry2017,9,204;doi:10.3390/sym9100204 www.mdpi.com/journal/symmetry
Symmetry2017,9,204 2of22
decision-making [42–44] and the theory of logical and analogical proportions [45–49]. In sum,
then,Aristoteliandiagramshavecometoserveasvisualtoolsthatgreatlyfacilitatecommunication,
researchandteachinginawidevarietyofdisciplinesthatdealwithlogicalreasoninginallitsfacets.
Figure1.Squareofoppositionforthecategoricalstatementsfromsyllogistics.
In the research program of logical geometry (see www.logicalgeometry.org), we study
Aristotelian diagrams as objects of independent mathematical interest, i.e., regardless of any of
their specific applications. We focus on logical issues such as informativity, Boolean complexity
andlogic-sensitivity[50–52],butalsoonmorevisual/diagrammaticaspects,suchasinformational
vs.computationalequivalenceofAristoteliandiagrams[53–55].Oneofthecrucialinsightsinthisareais
thatAristoteliandiagramscanalsobefruitfullyseenastrulygeometricalentitiesandstudiedbymeans
oftoolsandtechniquessuchasprojectionmatrices,Euclideandistance,symmetrygroups,etc.[54–57].
(Thishybridperspectiveondiagrams,treatingthemsimultaneouslyasdiagrammaticvisualizations
ofanunderlyingabstractstructureandasgeometricalentitiesbythemselves,canalsobefoundin
crystallography[58,59].) Theimportanceofthisgeometricalapproachisfurtherhighlightedbythe
followingobservation: althoughmostAristoteliandiagramsthathavebeenusedintheliteratureso
fararebasedonfairlysimple,two-dimensionalpolygonssuchassquares(cf.Figure1)andregular
hexagonsandoctagons, inrecentyears, variousresearchershavealsostartedusingmorevisually
complex,three-dimensionalAristoteliandiagrams,basedonpolyhedrasuchascubes,cuboctahedra
andrhombicdodecahedra[38,60,61].
The present paper further develops and refines this geometrical perspective on Aristotelian
diagrams. Inparticular,wewillconsiderfourthree-dimensional,polyhedralAristoteliandiagrams
B
that have recently been used to visualize the Boolean algebra , viz. the rhombic dodecahedron,
4
the tetrakis hexahedron, the tetraicosahedron, and the nested tetrahedron. (In his research on
paraconsistent logic, Béziau [62] has used various other Aristotelian diagrams, such as hexagons
andoctagons,butalsoanotherpolyhedron,viz.astellarrhombicdodecahedron. Thepreciselogical
andgeometricalrelationshipbetweenBéziau’spolyhedralAristoteliandiagramandtheonesdiscussed
inthispaperisdiscussedinmoredetailin[63].) Thiscasestudyisespeciallyrelevantinthecontext
ofartificialintelligence,sinceeachofthesefourvisualizationshasrecentlybeenusedinAI-related
research. In particular, the rhombic dodecahedron has been used in research on modal logic and
dynamic epistemic logic [22,61], the tetrakis hexahedron and tetraicosahedron have been used to
investigatemodallogic[25,60,64,65],andthenestedtetrahedronhasbeenusedtostudy(connections
between)theknowledgerepresentationformalismsofroughsettheory,formalconceptanalysisand
B
possibilitytheory[38,40]. (However,itshouldalsobenotedthatourfocuson precludesusfrom
4
studyingAristoteliandiagramsforknowledgerepresentationformalismsthatgobeyondaBoolean
Symmetry2017,9,204 3of22
setup,suchasfuzzylogicandsettheory[16–20,33–36]. Thesystematic(logicalanddiagrammatic)
investigationofsuchnon-classicalAristoteliandiagramsisamatterofongoingresearch.)
The paper makes two crucial new contributions to this line of research. First of all, it offers
adetailedinvestigationofthegeometricalpropertiesof,andinterrelationshipsbetween,thesefour
polyhedral Aristotelian diagrams. Some of these geometrical properties/interrelationships are
immediatelyclearfromthevisualfeaturesofthepolyhedra,butsomeofthemaremoreabstractin
nature(e.g.,inthecaseoftwopolyhedrathat,despitetheirclearvisualdifferences,canbeobtainedby
meansoftwodifferenttypesofprojectionsofoneandthesamehigher-dimensionalpolytope).Secondly,
itanalyzesthecorrespondence(orlackthereof)betweenthelogicalandgeometricalnotionsofdistance
ineachofthesefourpolyhedralAristoteliandiagrams. Logicaldistanceisdefinedintermsofthe
B
Hammingdistancebetweenthebitstringrepresentationsoftheelementsof ,whereasgeometrical
4
distanceisidentifiedwiththeEuclideandistancebetweeneachpolyhedron’sverticescorrespondingto
thosebitstringrepresentations. Theresultsofthiscomparativeanalysiswillthenbeusedtoshednew
lightontherelativestrengthsandweaknessesofthesepolyhedralAristoteliandiagrams,byappealing
tothecongruenceprinciplefromdiagramdesign[66,67].
As will be indicated throughout the remainder of the paper, there exists some earlier work
B
on polyhedral Aristotelian diagrams for . For example, Moretti [68] sketches the outlines of a
4
comparisonbetweentheAristoteliantetraicosahedronandnestedtetrahedron, butitgoesintofar
less geometrical detail than the present paper, and does not deal with the Aristotelian rhombic
dodecahedronandtetrakishexahedron. Similarly,Smessaertetal. [63,69]comparetheAristotelian
rhombicdodecahedron,tetrakishexahedronandtetraicosahedron,butthesestudiesagaingointofar
lessgeometricaldetailthanthepresentpaperanddonotaddresstheAristoteliannestedtetrahedron.
Finally,Smessaertetal. [55,57]offeranin-depthgeometricalcomparisonoftheAristotelianrhombic
B
dodecahedronandnestedtetrahedronfor ,butthesepapersdonotdealwiththeAristoteliantetrakis
4
hexahedronandtetraicosahedron.
B
Thepaperisorganizedasfollows. Section2brieflypresentstheBooleanalgebra ,itsbitstring
4
representationanditspolyhedralHassediagram. Section3,thenprovidesadetailedoverviewofthe
B
fourpolyhedralAristoteliandiagramsfor thathavebeenproposedintheliterature,focusingontheir
4
geometricalpropertiesandinterrelations,aswellasontheirlogicalapplications. Section4introduces
thenotionsoflogical(Hamming)distanceandgeometrical(Euclidean)distanceandanalyzeshow
closelythesetwonotionsofdistancearecorrelatedineachofthefourpolyhedralAristoteliandiagrams
B
for . Furthermore,theresultsofthiscomparativeanalysisareinterpretedinlightofthecongruence
4
principlefromcognitivestudiesondiagramdesign. Finally,Section5brieflysummarizestheresults
obtainedinthispaperandofferssomequestionsforfurtherresearch.
B
2. TheBooleanAlgebra andItsPolyhedralHasseDiagram
4
The notion of a Boolean algebra B = (cid:104)B,∧,∨,¬,⊥,(cid:62)(cid:105) is well-known in abstract algebra [70].
AsaspecialcaseoftheStonerepresentationtheorem,everyfiniteBooleanalgebraisisomorphictothe
powersetofsomefiniteset. Inparticular,theBooleanalgebraB isisomorphicto℘({x ,x ,x ,x }),
4 1 2 3 4
and elements of B can be represented as subsets of {x ,x ,x ,x } (the identity of the elements x
4 1 2 3 4 i
is irrelevant here). However, in logical geometry it is more customary to represent finite Boolean
B
algebrasbymeansofbitstrings[71]. Inparticular,theBooleanalgebra isrepresentedbymeansof
4
bitstringsoflength4,i.e.,B ∼= {0,1}4 (obviously,everybitstringoflength4uniquelycorresponds
4
to(thecharacteristicfunctionof)asubsetof{x ,x ,x ,x };forexample,thebitstring1010corresponds
1 2 3 4
tothesubset{x ,x }.)Thesebitstringscanbeusedtoprovideacoarse-grainedsemanticsforfragments
1 3
ofawidevarietyoflogicalsystems;see[51]forthemathematicaldetailsofthistechnique.
In a Boolean algebra B, the notion of level is inductively defined as follows: L := {⊥} and
0
Ln+1 := {b ∈ B | ∃a ∈ Ln: a < band¬∃c ∈ B: a < c < b}. In the bitstring representation of B4,
thelevelofabitstringsimplycorrespondstoitsnumberof1-bits;forexample0100∈ L and1011∈ L .
1 3
TheAristotelianrelationsareusuallydefinedforformulasofsomegivenlogicalsystem[50],butthis
Symmetry2017,9,204 4of22
can easily be generalized to arbitrary Boolean algebras [72,73]. In the specific case of the bitstring
representationofB ,wesaythattwobitstringsb ,b ∈ {0,1}4are:
4 1 2
contradictory(CD) iff b ∧b =0000 and b ∨b =1111,
1 2 1 2
contrary(C) iff b ∧b =0000 and b ∨b (cid:54)=1111,
1 2 1 2
subcontrary(SC) iff b ∧b (cid:54)=0000 and b ∨b =1111,
1 2 1 2
insubalternation(SA) iff b ∧b =b and b ∨b (cid:54)=b .
1 2 1 1 2 1
In the definition of CD, the condition that b ∧b = 0000 captures the informal idea that
1 2
contradictories‘cannotbetruetogether’,whiletheconditionthatb ∨b =1111capturestheinformal
1 2
ideathatthey‘cannotbefalsetogether’[72,73]. InthedefinitionsofCandSC,oneofthetwoidentity
conditionsofCDisnegated: contrariescanbefalsetogether,whilesubcontrariescanbetruetogether.
ThesetofAristotelianrelationsisfundamentallyhybridinnature: theoppositionrelationsCD,Cand
B
SCaredefinedusingthetopandbottomelementsof ,whiletheimplicationrelationSAisdefined
4
intermsofthebitstringsb andb themselves[50].
1 2
Since Boolean algebras are a particular kind of partially ordered sets, they can be visualized
bymeansofHassediagrams[74]. Thestandard,two-dimensionalHassediagramfor(thebitstring
B
representationof) isshowninFigure2a. Thisdiagramvisualizesnegationbymeansofcentral
4
symmetry: contradictory bitstrings are located at diametrically opposed vertices of the diagram.
Furthermore, in this Hasse diagram, the logical ordering of levels corresponds to the vertical
geometrical ordering of lines, from L at the bottom of the diagram up to L at the top. However,
0 4
several authors have also made use of polyhedra to provide alternative, three-dimensional Hasse
B B
diagramsfor [75,76]. Inparticular,Figure2bshowsaHassediagramfor basedonarhombic
4 4
dodecahedron.Justlikeitstwo-dimensionalcounterpart,thispolyhedralHassediagramalsovisualizes
negationbymeansofcentralsymmetry,whereasthelogicalorderingoflevelsnowcorrespondstothe
verticalgeometricalorderingofplanes,fromL atthebottomofthediagramuptoL atthetop.
0 4
Figure2. (a)Two-dimensionalHassediagramforB ;(b)Three-dimensionalHassediagramforB ,
4 4
basedonarhombicdodecahedron(withtheplanescorrespondingtothelogicallevelsshowningray).
Thegeometricalpropertiesoftherhombicdodecahedronwillbediscussedinmoredetailinthe
next section. (This geometrical information will play an important role in our discussion of the
(dis)similarities between the rhombic dodecahedron and the other polyhedra that are introduced
inSection3, andisthusmostnaturallyathomeinthatsection.) Fornow, itsufficestoemphasize
B
that the rhombic dodecahedron is a very natural choice for a polyhedral Hasse diagram for ,
4
because of a specific combination of logical and geometrical considerations. On the one hand,
B
the(bitstringrepresentationofthe)Booleanalgebra canalwaysberepresentedbymeansofan
n
n-dimensionalhypercube,withthelogicalbitvalues1/0correspondingtotheEuclideancoordinates
+1/−1[77](forexample,thebitstring10101inB correspondstothevertex(1,−1,1,−1,1)inR5).
5
Symmetry2017,9,204 5of22
B
Inparticular,theBooleanalgebra canthusberepresentedbymeansofafour-dimensionalhypercube,
4
i.e.,atesseract. Ontheotherhand,itiswell-knownthatthevertex-firstparallelprojectionofatesseract
fromfour-spaceintothree-spaceisexactlyarhombicdodecahedron[56]. Inparticular,therhombic
dodecahedroninFigure2bcanbeseenasthevertex-firstparallelprojectionofatesseractalongthe
projectionaxisdefinedby(theverticescorrespondingto)thebitstrings1001and0110,becausethose
arepreciselythetwobitstringsthatendupinthecenterofpolyhedron,ratherthanonitsexterior
surface. Actually,inordertoavoidthat1001and0110entirelycoincidewitheachotherinthecenter
of the polyhedron, a slight distortion in the projection axis has been introduced; this procedure is
standard[78],andtheHasserhombicdodecahedroncanthereforebesaidtobea‘quasi-vertex-first
projection’ofatesseract[56].
B
3. PolyhedralAristotelianDiagramsfor
4
B
Intheprevioussection,wehaveintroducedtheBooleanalgebra andaspecificpolyhedral
4
Hassediagramforit. Intheremainderofthispaper, however, wewillratherfocusonpolyhedral
B B
Aristotelian diagrams for . The latter emphasize different logical aspects of ; in particular,
4 4
B
theyfocusonclearlyvisualizingtheAristotelianrelationsholdingamongtheelementsof . Inorder
4
toachievethisgoal,otherlogicalinformationisdistortedorleftoutaltogether;forexample,thetop
B
andbottomelementsof ,i.e.,thenon-contingentbitstrings1111and0000,arenotshownatallinan
4
Aristoteliandiagram(becausetheyenterintoabundantlymany‘vacuous’Aristotelianrelations[50]),
andthelogicalorderingoflevelsisnolongerreflectedinaverticalorderingofplanes.
B
AtleastfourdifferentpolyhedralAristoteliandiagramsfor havebeenproposedintheliterature.
4
B
Thesediagramsallrepresentthesamelogicalstructure(viz. ),buttheyachievethisbymeansof
4
very different visual means. In Larkin and Simon’s terminology [79], these Aristotelian diagrams
areinformationallyequivalent,buttheyneednotbecomputationallyequivalent,inthesensethat
thevisualdifferencesbetweenthemmaysignificantlyinfluenceusercomprehension. Inthissection,
wewillprovideanin-depthcomparativeoverviewofthesefourpolyhedralAristoteliandiagrams
B
for .
4
B
3.1. TheAristotelianRhombicDodecahedronfor
4
B
We start our overview by considering the Aristotelian rhombic dodecahedron (RDH) for ,
4
asshowninFigure3a. ThisAristoteliandiagramwasfirstproposedbySmessaert[61]inthecontext
ofhisresearchongeneralizedquantifiersandmodallogicandlateradoptedbyDemey[22]inhis
workonpublicannouncementlogic. TheHassediagramdiscussedinSection2andtheAristotelian
diagramconsideredherearebothbasedonarhombicdodecahedron; theonlydifferencebetween
B
these diagrams concerns how the bitstrings of are mapped onto the vertices of these rhombic
4
dodecahedra;compareFigures2band3a.
Symmetry2017,9,204 6of22
Figure3.(a)AristotelianrhombicdodecahedronforB ;(b)Therhombicdodecahedronasacubewith
4
pyramidsputontoeachofitssixfaces.
The rhombic dodecahedron has 14 vertices, 24 edges and 12 (rhombic) faces; being a convex
polyhedron,ithasEulercharacteristicχ = 2,i.e.,V−E+F = 2. Notethatthenumberofvertices
(V =14)correspondsexactlytothenumberofcontingentbitstringsofB : thisBooleanalgebrahas
4
24 =16bitstringsintotal,soafterdiscarding1111and0000,weareleftwith14contingentbitstrings.
TherhombicdodecahedronisaCatalansolid[80],andhasasitssymmetrygrouptheoctahedralgroup
O (oforder48),whichitshareswithitsdualpolyhedron,thecuboctahedron[63,81,82]. Thelatter
h
is itself an Archimedean solid and has also been used as a polyhedral Aristotelian diagram [60].
Furthermore,therhombicdodecahedroniscloselyrelatedtoboththecubeandtheoctahedron(which
arethemselvesdualtoeachother),asshowninFigure4[63,69,83].
Figure4.Therhombicdodecahedronasthecompoundofacubeandanoctahedron.
B
JustliketheHasserhombicdodecahedronfor discussedintheprevioussection,theAristotelian
4
B
rhombic dodecahedron for is also the vertex-first parallel projection of a tesseract, but now
4
along the projection axis defined by (the vertices corresponding to) the bitstrings 1111 and 0000.
Furthermore,sincethenon-contingentbitstrings1111and0000areirrelevantinAristoteliandiagrams,
itisunproblematicthattheyexactlycoincidewitheachotherinthecenterofthepolyhedron,andhence,
thereisnoneedtoinvokeaquasi-vertex-firstprojection.ByviewingtheHasseandAristotelianrhombic
dodecahedraastwovertex-firstparallelprojectionsofatesseract,albeitalongdifferentprojectionaxes,
wealsoobtainaunifiedgeometricalexplanationoftheirfundamentaldiagrammaticdifferences[56].
In particular, in the tesseract, the logical levels are ordered by means of hyperplanes going from
Symmetry2017,9,204 7of22
(thevertexcorrespondingto)0000to(thevertexcorrespondingto)1111;however,iftheprojection
ispreciselyalongthe1111/0000axis,thenthisorderingiscompletelyannihilated,sothattheresulting
Aristotelianrhombicdodecahedronnolongervisuallyrepresentsthisorderingoflevels: theL -,L -
1 2
andL -bitstringsarescatteredthroughoutthepolyhedron[55,57].
3
As is illustrated in Figure 3b, the rhombic dodecahedron can be seen as the result of putting
asquarepyramidontoeachofthefacesofacube[84]. Theheightofthesesixpyramidsshouldbe
suchthatthelateral,triangularfacesofadjacentpyramidslieinthesameplaneandtogetherform
asingle,rhombicfaceoftherhombicdodecahedron. Itiseasytoshowthatifthecubehasedgelength
(cid:96),thiscanbeachievedbyusingpyramidsofheight (cid:96). Inthesepyramids,thebaseisasquareofedge
2
length(cid:96)(viz.oneofthefacesofthecube),andthelateralfacesareisoscelestriangles(withoneedge
√
oflength(cid:96)andtwooflength 3(cid:96));thelateralfacesareatanangleof45◦ tothebase. Thisperspective
2
ontherhombicdodecahedronwillbeusefullateron,whenwearecomparingitwithotherpolyhedra.
ItwillbeconvenienttofixasetofstandardcoordinatesfortheverticesoftheAristotelianrhombic
dodecahedron. Thecoordinatefunctionc : B → R3 showninTable1attheendofthissection
RDH 4
B
mapsthebitstringsof ontotheverticesoftheAristotelianrhombicdodecahedron[55,57]. Notethat
4
c (1111) = c (0000) = (0,0,0)andc (¬b) = −c (b)forallbitstringsb,i.e.,thetopand
RDH RDH RDH RDH
B
bottomelementsof coincidewitheachotherinthecenteroftheAristotelianrhombicdodecahedron,
4
andlogicalnegationisvisualizedbymeansofcentralsymmetry. TheL -andL -bitstringsaremapped
1 3
ontothevertices (±1,±1,±1) andconstitutetwoequal-sizedtetrahedrathat‘interlock’witheach
other to yield a cube, whereas the L -bitstrings are mapped onto the vertices (±2,0,0), (0,±2,0)
2
and(0,0,±2),whichconstituteanoctahedron;cf.Figure5. (Ofcourse,thevertices(±2,0,0),(0,±2,0)
and(0,0,±2)canalsobeseenasthetopsofthepyramidsthathavebeenputoneachofthecube’ssix
faces;cf.Figure3b.) Notethatthecubehasedgelengthtwo,andhencethepyramidsthatareputon
topofthecube’sfaceshaveheightone;forexample,thetopfaceofthecubeliesintheplaney = 1;
theapexofthepyramidthatisputontopofthisface,liesat(0,2,0).
Table1.CoordinatefunctionsforthefourpolyhedralAristoteliandiagramsforB .
4
b∈B c (b) c (b) c (b) c (b)
4 RDH THH TIH NTH
0000 (0,0,0) (0,0,0) (0,0,0) (0,0,0)
1000 (−1,1,−1) (−1,1,−1) (−1,1,−1) (−1,1,−1)
0100 (−1,−1,1) (−1,−1,1) (−1,−1,1) (−1,−1,1)
0010 (1,−1,−1) (1,−1,−1) (1,−1,−1) (1,−1,−1)
0001 (1,1,1) (1,1,1) (1,1,1) (1,1,1)
√
1100 (−2,0,0) (−3,0,0) (−1− 2,0,0) (−1,0,0)
2 √
1010 (0,0,−2) (0,0,−3) (0,0,−1− 2) (0,0,−1)
2 √
1001 (0,2,0) (0,3,0) (0,1+ 2,0) (0,1,0)
2 √
0110 (0,−2,0) (0,−3,0) (0,−1− 2,0) (0,−1,0)
2 √
0101 (0,0,2) (0,0,3) (0,0,1+ 2) (0,0,1)
2 √
0011 (2,0,0) (3,0,0) (1+ 2,0,0) (1,0,0)
2
1110 (−1,−1,−1) (−1,−1,−1) (−1,−1,−1) (−1,−1,−1)
3 3 3
1101 (−1,1,1) (−1,1,1) (−1,1,1) (−1,1,1)
3 3 3
1011 (1,1,−1) (1,1,−1) (1,1,−1) (1,1,−1)
3 3 3
0111 (1,−1,1) (1,−1,1) (1,−1,1) (1,−1,1)
3 3 3
1111 (0,0,0) (0,0,0) (0,0,0) (0,0,0)
Symmetry2017,9,204 8of22
Figure5.(a)TetrahedronfortheL -bitstrings;(b)octahedronfortheL -bitstringsand(c)tetrahedron
1 2
fortheL -bitstringsinsidetheAristotelianrhombicdodecahedronforB .
3 4
B
3.2. TheAristotelianTetrakisHexahedronfor
4
We now turn to the Aristotelian tetrakis hexahedron (THH) (or simply: tetrahexahedron) for
B
,whichisshowninFigure6a. ThisdiagramwasfirstusedbySauriol[85]inhisresearchonthe
4
logicalgeometryofthepropositionalconnectivesandwaslateralsoadoptedbyLuzeauxetal.[25]
andPellissier[64]intheirresearchonmodallogic. (Althoughtheselatterauthors[25,64]erroneously
calltheirdiagrama‘tetraicosahedron’;cf.infra.) Thetetrakishexahedronhas14vertices,36edges
and24faces;beingaconvexpolyhedron,itsatisfiestheEulerformulaV−E+F =2. Notethatthe
numberofvertices(V =14)againcorrespondsexactlytothenumberofcontingentbitstringsofB
4
(cf.supra). ThetetrakishexahedronisagainaCatalansolid;itisdualtothetruncatedoctahedron,
whichisitselfagainanArchimedeansolid;thesymmetrygroupofboththesepolyhedraisagainthe
octahedralgroupO [80–82].
h
Figure 6. (a) Aristotelian tetrakis hexahedron for B ; (b) The tetrakis hexahedron as a cube with
4
pyramidsputontoeachofitssixfaces.
Just like the rhombic dodecahedron, the tetrakis hexahedron can be seen as the result of
putting a square pyramid onto each of the faces of a cube; cf. Figure 6b (this was also noted by
Sauriol[85],pp.384–385). Ifthecubehasedgelength(cid:96),thepyramidshaveheight (cid:96). Inthesepyramids,
4
thebaseisasquareofedgelength(cid:96)(viz.oneofthefacesofthecube),andthelateralfacesareisosceles
triangles(withoneedgeoflength(cid:96)andtwooflength 3(cid:96)). Comparingtherhombicdodecahedronin
4
Figure3aandthetetrakishexahedroninFigure6a,wefindthatthesetwoAristoteliandiagramsare
verysimilartoeachother. Thecrucialdifferenceconcernstheheightofthesixpyramids,whichis
(cid:96)
muchsmallerinthelatterthanintheformer( inthepyramidsofthetetrakishexahedronversus
4
Symmetry2017,9,204 9of22
(cid:96)
inthoseoftherhombicdodecahedron). Consequently,inthetetrakishexahedron,thepyramids’
2
lateralfacesareatanangleofstrictlylessthan45◦ tothebase,andhencethelateralfacesofadjacent
pyramidsnolongerlieinthesameplane,andthusnolongerformasingle,rhombicface. Takingthe
rhombicdodecahedronasourstartingpoint, eachofthe12rhombicfacesisthusbrokenintotwo
triangularfaces,whichexplainswhythetetrakishexahedronhastwiceasmanyfacesastherhombic
dodecahedron(24versus12)andhas12additionaledges(36versus24).
WeagainfixasetofstandardcoordinatesfortheverticesoftheAristoteliantetrakishexahedron.
Thecoordinatefunctionc : B →R3showninTable1attheendofthissectionmapsthebitstrings
THH 4
B
of onto the vertices of the Aristotelian tetrakis hexahedron. This coordinate function reveals
4
the strong similarities between the Aristotelian rhombic dodecahedron and tetrakis hexahedron.
Firstofall,notethatweagainhavec (1111) = c (0000) = (0,0,0)andc (¬b) = −c (b)
THH THH THH THH
B
for all bitstrings b, i.e., the top and bottom elements of coincide with each other in the center
4
oftheAristoteliantetrakishexahedron,andlogicalnegationisvisualizedbymeansofcentralsymmetry.
Furthermore,theL -andL -bitstringsareonceagainmappedontothevertices(±1,±1,±1)ofthe
1 3
cube. The only difference between c and c concerns the coordinates for the L -bitstrings,
RDH THH 2
i.e.,theheightofthesixpyramidsthatareputontopofthecube’sfaces. Sincethecubehasedge
length two, the pyramids that yield the tetrakis hexahedron have height 1 ×2 = 1; for example,
4 2
thetopfaceofthecubeliesintheplaney =1;theapexofthepyramidthatisputontopofthisface,
liesat(0,1+ 1,0) = (0,3,0). Ingeneral,itholdsthatc (b) = 3c (b)forallb ∈ L ,whichreflects
2 2 THH 4 RDH 2
thefactthatinthetetrakishexahedron,thepyramids’apicesareclosertothecube’sfacesthanwasthe
caseintherhombicdodecahedron.
B
3.3. TheAristotelianTetraicosahedronfor
4
B
Thenextpolyhedrontobeconsidered,istheAristoteliantetraicosahedron(TIH)for ,whichis
4
showninFigure7a,andwhichwasfirstusedbyMoretti[60,65]inhisresearchonn-oppositiontheory
andmodallogic. (Asfarasweknow,thispolyhedronisnotsystematicallystudiedinthegeometrical
literatureonpolyhedra;theterm‘tetraicosahedron’onlyseemstooccurinthelogic-orientedresearch
ofMoretti,PellissierandLuzeauxetal. Furthermore,thistermhasnotbeenusedentirelyconsistently
in the literature; for example, Pellissier [64] and Luzeaux et al. [25] draw a tetrakis hexahedron,
butcallita‘tetraicosahedron’,whileMoretti[68],conversely,drawsatetraicosahedron,butcallsit
a‘tetrahexahedron’.) Justlikethetetrakishexahedron,thetetraicosahedronhas14vertices,36edges
and 24 faces; even though this polyhedron is not convex, it thus still satisfies the Euler formula
V−E+F =2. Thetetraicosahedrondoesnotbelongtoanyofthewell-studiedfamiliesofpolyhedra
(cf.supra). Itiseasytosee,however,thatitssymmetrygroupis,onceagain,theoctahedralgroupO .
h
Furthermore,thenumberofvertices(V =14)againcorrespondsexactlytothenumberofcontingent
B
bitstringsof (cf.supra).
4
Onceagain,thetetraicosahedroncanbeseenastheresultofputtingasquarepyramidontoeach
ofthefacesofacube;cf.Figure7b. However,theideaisnowthatthelateralfacesofthesepyramids
shouldbeequilateral(ratherthanmerelyisosceles)triangles. Itiseasytoshowthatifthecubehas
edgelength(cid:96),thiscanbeachievedbyusingpyramidsofheight √(cid:96) . Insuchpyramids,thebaseisa
2
squareofedgelength(cid:96),andthelateralfacesareequilateraltrianglesofedgelength(cid:96);thesepyramids
areJohnsonsolids[86]. ComparingtherhombicdodecahedroninFigure3aandthetetraicosahedron
inFigure7a,wefindthatthesetwoAristoteliandiagramsare,onceagain,verysimilartoeachother.
The crucial difference concerns the height of the six pyramids, which is much greater in the latter
than in the former (√(cid:96) in the pyramids of the tetraicosahedron versus (cid:96) in those of the rhombic
2 2
dodecahedron). Consequently,inthetetraicosahedron,thepyramids’lateralfacesareatanangleof
strictlymorethan45◦ tothebase,andhencethelateralfacesofadjacentpyramidsnolongerlieinthe
sameplane,andthusnolongerformasingle,rhombicface. Takingtherhombicdodecahedronasour
startingpoint,eachofthe12rhombicfacesisthusbrokenintotwotriangularfaces,whichexplains
whythetetraicosahedronhastwiceasmanyfacesastherhombicdodecahedron(24versus12),andhas
Symmetry2017,9,204 10of22
12additionaledges(36versus24). Finally,anotherconsequenceofthepyramids’greaterheightsis
thatthetetraicosahedronisnotconvex;forexample,thebitstrings1100and1001areontwoverticesof
thetetraicosahedron,butthelinesegmentbetweenthesetwoverticesclearlyliesentirelyoutsidethis
polyhedron;cf.Figure7a.
Figure7.(a)AristoteliantetraicosahedronforB ;(b)Thetetraicosahedronasacubewithpyramids
4
putontoeachofitssixfaces.
ItwillagainbeconvenienttofixasetofstandardcoordinatesfortheverticesoftheAristotelian
tetraicosahedron. Thecoordinatefunctionc : B →R3showninTable1attheendofthissection
TIH 4
B
mapsthebitstringsof ontotheverticesoftheAristoteliantetraicosahedron.Thiscoordinatefunction
4
revealsthestrongsimilaritiesbetweentheAristotelianrhombicdodecahedronandtetraicosahedron.
First of all, note that we again have c (1111) = c (0000) = (0,0,0) and c (¬b) = −c (b)
TIH TIH TIH TIH
B
for all bitstrings b, i.e., the top and bottom elements of coincide with each other in the center
4
oftheAristoteliantetraicosahedron,andlogicalnegationisvisualizedbymeansofcentralsymmetry.
Furthermore,theL -andL -bitstringsareonceagainmappedontothevertices(±1,±1,±1)ofthe
1 3
cube. The only difference between c and c concerns the coordinates for the L -bitstrings,
RDH TIH 2
i.e.,theheightofthesixpyramidsthatareputontopofthecube’sfaces. Sincethecubehasedge
√
length two, the pyramids that yield the tetraicosahedron have height √1 ×2 = 2; for example,
2
thetopfaceofthecubeliesintheplaney =1;theapexofthepyramidthatisputontopofthisface,
√ √
liesat(0,1+ 2,0). Ingeneral,itholdsthatc (b) = 1+ 2c (b)forallb ∈ L ,whichreflectsthe
TIH 2 RDH 2
factthatinthetetraicosahedron,thepyramids’apicesarefartherremovedfromthecube’sfacesthan
wasthecaseintherhombicdodecahedron.
B
3.4. TheAristotelianNestedTetrahedronfor
4
B
The final polyhedron to be considered, is the Aristotelian nested tetrahedron (NTH) for ,
4
asshowninFigure8. Ciucci,DuboisandPradehaverecentlymadeuseofthisAristoteliandiagramin
theircomparativeresearchonthelogicalgeometryofroughsettheory,formalconceptanalysisand
possibilitytheory[38,40]. Furthermore,recenthistoricalresearchhasshownthatapredecessorofthis
diagramwasalreadyusedbythe19th-centurylogician(andnovelist)LewisCarroll[68,87].