The spatial distribution of the Lyα forest is studied using new HST data for the quasar pair Q~1026--0045 A and B at zem = 1.438 and 1.520 respectively. The angular separation is 36 arcsec and corresponds to transverse linear separations between lines of sight of ∼300h50−1 kpc (qo = 0.5) over the redshift range 0.833 < z < 1.438. From the observed numbers of coincident and anti-coincident Lyα absorption lines, we conclude that, at this redshift, the Lyα structures have typical dimensions of ∼500h50−1 kpc, larger than the mean separation of the two lines of sight. The velocity difference, ΔV, between coincident lines is surprisingly small (4 and 8 pairs with ΔV<50and200km/srespectively).Metallinesystemsarepresentatz_{abs}=1.2651and1.2969inA,z_{abs}=0.6320,0.7090,1.2651and1.4844inB.InadditionwetentativelyidentifyaweakMgIIsystematz_{abs}=0.11inB.Itisremarkablethatthez_{abs}=1.2651systemiscommontobothlinesofsight.Thesystematz_{abs}=1.4844z_{abs}=1.4420alongthelineofsighttoAwithcomplexvelocityprofile.WedetectastrongLy\alphaabsorptionalongthelineofsighttoBredshiftedbyonly300km/srelativelytotheassociatedsystem.Itistemptingtointerpretthisasthepresenceofadiskofradiuslargerthan300h^{-1}_{50}$ kpc surrounding quasar A.