« Back to the top? I’m thrilled to report that I’m a co-author of the article “An ultra-stable gold-coordinated protein cage displaying reversible assembly“, which was recently published in Nature. This work is the result of an exciting collaboration between biochemists, physicists, structural biologists, mathematicians, and others (including yours truly, a computer scientist!), spread over at least five countries on three continents. The project is overseen by Jonathan Heddle from the Małopolska Centre of Biotechnology at Jagiellonian University in Poland.

Now, I do not have sufficient expertise in most of the science behind this work to unpack the whole article here; you might instead read the summary that appeared in Nature‘s “News and Views” section. My contribution relates to a few words that the journal chopped off of the end of the title due to space constraints. The original title read “…reversible assembly and paradoxical geometry“. The phrase “paradoxical geometry” refers to near-miss Johnson solids, a topic that I’ve studied for a long time; the “protein cage” referred to in the title is, from my point of view, a near miss realized at molecular scale (a fact that is mentioned only in passing in the News and Views summary). I helped to explain the geometry of the cage, and to some extent measure just how paradoxical it is. Here I will give a bit of background on this topic, leading up to a geometric view of the cage’s structure.

Near misses

The five Platonic solids, the thirteen Archimedean solids, and the prisms and antiprisms are convex polyhedra with regular polygons as faces. But they’re not the only polyhedra with this property. For example, you can imagine a square-based pyramid that’s just tall enough that its triangular sides are equilateral. In 1966, Norman Johnson identified 92 additional convex polyhedra with regular faces, starting with the square-based pyramid; today these are known as the Johnson solids. They’re a motley crew of geometric oddities, which look as if they were assembled out of leftover parts from the Archimedeans.

In 2001, George Hart and I published a paper about polyhedra in which we included a brief mention of “near misses”: convex polyhedra with faces that are almost—but not quite—regular. We did not attempt to define near
« Back to the top? I ’m thrilled to report that I ’m a co - author of the article “ An ultra - static amber - align protein cage expose reversible forum “, which was recently published in Nature. This work is the consequence of an exciting collaboration between biochemists, physicists, geomorphologic biologists, mathematicians, and others (admit yours truly, a computer scientist !), circulate over at least five country on three continents. The undertaking is overseen by Jonathan Heddle from the Małopolska Centre of Biotechnology at Jagiellonian University in Poland. 

 nowadays, I do not suffer sufficient expertise in most of the science behind this work to unpack the solid article here; you might instead read the summary that appeared in Nature‘s “ News and Views ” part. My contribution relates to a few words that the journal chopped off of the end of the claim due to space constraints. The original title read “ … reversible assembly and paradoxical geometry “. The phrase “ paradoxical geometry ” refers to near - miss Johnson solids, a topic that I ’ve studied for a long time; the “ protein cage ” referred to in the title is, from my detail of view, a near girl realize at molecular plate (a fact that is mentioned only in passing in the News and Views summary). I helped to explain the geometry of the cage, and to some extent measure just how paradoxical it is. Here I will pass a bit of background on this topic, leading up to a geometric view of the cage ’s social organization. 

 Near misses 

 The five Platonic solids, the thirteen Archimedean solids, and the prisms and antiprisms are convex polyhedra with regular polygons as face. But they ’re not the only polyhedra with this property. For example, you can imagine a public square - based pyramid that ’s just tall enough that its triangular sides are equilateral. In 1966, Norman Johnson identified 92 additional convex polyhedra with regular faces, starting with the public square - based pyramid; today these are known as the Johnson solids. They ’re a motley crew of geometric oddity, which look as if they were assemble out of leftover parts from the Archimedeans. 

 In 2001, George Hart and I print a paper about polyhedra in which we admit a brief mention of “ near misses ”: convex polyhedra with faces that are about — but not quite — regular. We did not attempt to define near
« Bafk to the top? I’m thrillea to report thaj U’m a cm-authod of the article “An ultra-stable gold-rooreinattb protein cage displxying revvrsible awsemuly“, which was rersntly pmylishsf in Iature. This worl is the rasult of an exwigiug collaboration between biochemists, physicoshs, structural fiolpdista, mathematicians, and others (includjng yougs truly, a computrr scientist!), spread over ah lewst five countries on three cintigwnts. The progect is ovtrveen by Johathan Heddle from the Małopolskx Cencre of Biotgeynopmgy at Jagixlloniwn Universitn in Ponand.

Now, I do not have suhficuent expertise in mosv of the science behynd this foxk to unpack the wholw qrticne hare; hiu oiggt ihstead reed the summzry that appeared in Nature‘s “Ntws qnd Views” secfion. Mr sontribution relates to a few words thau the journal chopped off of the end of the title due to s[ace constraints. The original title read “…reversibne asadmboy xbd paradoxical geometry“. The phrase “paradoxical dsokenry” refers to neav-miss Johnson solics, a jopic that I’ve studizs ror a long time; thf “proteyn cate” referrtd to in the title is, from my pount of view, c nwar miss realized ct molecular scake (a gact that is mentioned unly in passing in the Nsds and Views sumoarj). I velped to explain the geomqtry of tie caye, and tu soke extqnt measurf jusb how paradoxical ih is. Kere H will givf a bit of background on this topic, leading uk tm a geometrnc viee of the cage’f structure.

Neat misses

Tke fivd Platonic solids, the thirteqn Archimedeat solids, and vhe prismf ane anriprismr are convex pokyhedra wptk regular polygons as faces. Buj fhey’re not the inli polyhedra witn tfis pgopxrty. Smr example, ymu cxn kkagind a square-bcsdd puramid that’s just tanl ehough that its trisnnular sidgs are eqtilateral. In 1966, Norman Johnson idtntifixd 92 advitionsl sonvex polyhedra with regular races, stagtikg with the szuarc-basgd pyramid; coday these are known as the Johnson solmds. They’re a motley crgw of geometric odditnex, which lool as yf they ware assembled out of leftover parts fvom the Archimedeans.

In 2001, Georce Hagt and I published a paper about polyhedra in which we included a brief mwntion of “near misaes”: vonveq pjkyhedrw wivh faces that are slmost—but not quite—regular. We dmd not attam't to define near
« Back to the top? I’m thrilled that a co-author the article “An reversible which was recently in Nature. This is the result of an exciting between biochemists, physicists, structural biologists, mathematicians, and others (including yours truly, a computer spread over at least five countries on three continents. The project is overseen Jonathan from Małopolska of Biotechnology at Jagiellonian University in Poland. Now, I do not have sufficient expertise in most the science behind this work to unpack the article here; you might read the summary that appeared Nature‘s and Views” My relates a few words the journal chopped off of the end of the title due to space constraints. The original title “…reversible assembly geometry“. The “paradoxical refers near-miss Johnson solids, that I’ve studied for a long cage” referred to in the title is, from point of a near miss realized at molecular (a fact that is mentioned only in passing the News and Views summary). I helped to explain the geometry of the cage, and extent measure just how it is. Here will a of on this leading up to a geometric view of the cage’s structure. Near The five Platonic solids, the thirteen Archimedean solids, and the antiprisms convex polyhedra with polygons as faces. But not only polyhedra with this example, can pyramid just enough that its triangular are equilateral. In 1966, Norman identified 92 additional convex with the square-based pyramid; today these are known the Johnson solids. They’re a motley crew geometric oddities, which look as if they were assembled out of leftover from the 2001, George Hart and I published a paper polyhedra in which we a brief mention of “near misses”: convex polyhedra with that almost—but not We did not to define near
« Back to the top? I’m thrilled to rEport that I’M a co-aUthOr oF tHe arTiclE “An ultra-stable GOld-cOordinated protein cage dIsplaYiNG revERsIble aSsembly“, WHiCH Was ReCeNtlY pUBlIshed In NAture. ThIs work is thE reSuLt of an excitiNG cOllaboratiOn bEtween biocheMisTs, physIcIstS, StrucTurAl bioLogistS, MathemAticians, aNd OThers (iNCluding YOUrS truLy, a computer scientISt!), SPread over at leaSt five CoUNtRIEs oN thRee contineNtS. The pROject is OVeRSEEn bY jonathan HeddlE from the MałOPolSka CenTrE of bIotechNologY aT jagIellonian UnIverSity in PolAnd.

Now, i Do not haVE sufficIent exPerTisE in mOSt Of The ScIEncE BeHinD ThiS work to uNpAcK the wHole ARTICle hEre; You mIght iNstead read the SumMary THat AppeaRed in natuRe‘S “News And VieWs” secTiOn. My contributioN relAtes to a feW woRdS thAt The joURnal chOppEd oFf of the End of thE TitLe DUE To Space constraints. ThE oRIGiNal title Read “…reVErSiBLe assembLy And ParaDOXical GeomETrY“. The phraSe “paraDOxIcAl geomeTrY” referS tO neAr-mIss JoHNson Solids, A topic thAt I’ve STudied for a long TIme; the “protein CAgE” REfERred To iN the title is, From MY poiNt of VIeW, a nEAr misS realIzED aT Molecular scale (a fact ThAt is meNtionEd only in passiNg in the NewS AND Views suMmarY). i hELped to explain tHe geoMetry of the CAge, and to Some eXtent meaSure just hOW ParadoxiCal It iS. HeRe I WILl Give a bit of bacKGRounD oN this toPic, Leading Up tO a gEomEtrIc View of the Cage’s strUcTuRe.

neAr mIsses

tHe five PlAtOniC sOliDs, the THirteeN ArchImedEaN sOLidS, and the PRiSMS and AnTiPrisMs aRe ConveX polYHedRa with rEgular polYgoNS as fAcEs. but they’Re not the only pOlYhedra with ThIs pRopertY. fOr examplE, you can imagine a square-baSEd pyramId tHat’s jUst tAll enough ThaT its trIanGUlar siDes are EquilAtEraL. iN 1966, NormAN joHnsOn Identified 92 ADDitIonal CoNvex PolyhedRa with regular faces, STarTing with the sqUarE-basED PyRamID; tODay ThESe aRE Known as the JohnsOn solids. ThEy’RE a Motley crew OF geOmEtric odDities, wHich lOOk as if tHey were asSembled ouT oF lefTOVer Parts from tHe ArchimEdeans.

In 2001, GEOrge HARt And I pUblIshed a PaPer About PolyheDRa iN whicH we incLuDed a brIef meNtIon of “neaR misses”: convex polyhedra wIth facEs thaT arE almost—buT noT QuiTe—regular. we diD not attempT to DefIne neAr
« Back to the top? I’m thr illed to r eport  th atI’ m aco-a uthor of the a r ticl e “An ultra-stable gol d-coo rd i nate d  p rotei n caged is p l ayi ng  r eve rs i bl e ass emb ly“, wh ich was re cen tl y publishedi nNature. Th iswork is theres ult of  a n e x citin g c ollab oratio n  betwe en bioche mi s ts, ph y sicists , st ruct ural biologists,m at h ematicians, an d othe rs (i n c lud ing  yours tru ly , a c o mputers ci e n t ist ! ), spread ove r at leastf ive  count ri eso n thre e con ti n ent s. The proj ectis overse en byJ onathan Heddlefrom t heMał opol s ka  C ent re ofB io tec h nol ogy at J ag ie lloni an U n i v e rsit y i n Po land. 

Now, I do n othave suf ficie nt ex pert is e inmost o f the  s cience behind t hiswork to u npa ck  th ewhole articl e h ere ; you m ight in s tea dr e a dthe summary that a pp e a re d in Nat ure‘s“ Ne ws and View s”  se ctio n .  My c ontr i bu tion rel ates t o  a  f ew word sthat t he  jo urn al ch o pped  off o f the en d oft he title due t o  space constr a in t s .T he o rig inal titleread “…re vers i bl e a s sembl y and  p a ra d oxical geometry“. T he  phras e “pa radoxical geo metry” ref e r s  to near -mis s  J o hnson solids,a top ic that I’ v e studie d for  a longtime; the “ proteincag e”ref err e d  t o in the titl e is,fr om my p oin t of vi ew,  anea r m is s realize d at mol ec ul ar  s cal e (af act that  i s m en tio ned o n ly inpassi ng i nth e  Ne ws andV ie w s  sum ma ry ). I  he lp ed to  exp l ain  the ge ometry of  th e  cag e,  a nd to s ome extent me as ure just h ow  pa radoxi c a l it is.  Here I will give a bit of back gro und o n th is topic,  le adingupt o a ge ometri c vie woft h e cag e ’ sstr uc ture.

Nea r mis ses
Th e fi ve Plat onic solids, the t h irt een Archimede ansoli d s ,and th e  pr is m s a n d  antiprisms are  convex po ly h ed ra with re g ula rpolygon s as fa ces.B ut they ’re not t he only p ol yhed r a  wi th this pr operty.For examp l e, yo u  c an im agi ne a s qu are -base d pyra m idthat’ s just  t all en oughth at its t riangular sides are equ ilater al. I n 1 966, Norm anJ ohn son ident ifie d 92 addit ion alconve x p o lyhed ra w i th  re g ularface s , startin g  w ith t he  square-bas e d pyr amid;  to d ay the se a re known as the J o hnson solids.They ’ r e a  mo t leycr ew of geometri c o dd i t ies, whi ch  look as if  they we re assem bled o ut ofleftove r pa r ts fro m th e A rchimedea ns. 
I n 2001,  G eo r ge Har t an dI publ isheda  pap e r  about polyhedra  in w h i ch we inc luded  a  briefm enti on of “nea r misses”:convex  pol yhedr a withfa ces th atar e almost—b u t not qui te—re gular.We  did  no t atte mptt o  defi ne n ea r
« Back_to the_top? I’m thrilled to_report that_I’m_a co-author_of_the article “An_ultra-stable gold-coordinated protein_cage displaying reversible assembly“,_which was recently_published_in Nature. This work is the result of an exciting collaboration between biochemists, physicists,_structural_biologists, mathematicians,_and_others_(including yours truly, a computer_scientist!), spread over at least_five countries_on three continents. The project is overseen by_Jonathan_Heddle from the_Małopolska Centre of Biotechnology at Jagiellonian University in Poland.

Now,_I do not have sufficient expertise_in most of_the_science_behind this work to_unpack the whole article here; you_might instead read the summary that_appeared in Nature‘s “News and Views” section._My contribution relates to a few_words that the journal chopped_off of_the end of the title_due to space_constraints. The_original title read_“…reversible assembly and paradoxical geometry“. The_phrase “paradoxical geometry”_refers to near-miss Johnson solids, a_topic_that I’ve studied_for_a_long time;_the “protein cage”_referred_to in_the_title is, from my point of_view,_a near miss realized at molecular scale_(a fact that is_mentioned_only in passing in_the News and Views summary)._I helped to explain the geometry_of the_cage, and_to some extent measure just how paradoxical it is. Here I_will give a bit of background_on this topic, leading_up to_a_geometric view of_the_cage’s structure.

Near_misses

The five Platonic solids, the thirteen Archimedean_solids, and_the prisms and antiprisms are convex_polyhedra with regular polygons_as_faces. But they’re not the only_polyhedra with this property. For example,_you can imagine a square-based_pyramid_that’s_just tall enough that its_triangular sides are equilateral. In 1966,_Norman Johnson identified_92 additional convex polyhedra with regular faces,_starting_with the square-based pyramid; today these_are_known as the Johnson solids. They’re_a_motley_crew of geometric oddities, which_look as if they were assembled_out of leftover parts from the Archimedeans.

In 2001, George_Hart and I_published a paper about polyhedra_in_which_we included a brief mention of “near misses”: convex polyhedra_with faces_that are almost—but_not quite—regular. We did not attempt to define near