The torus can also be described as a quotient of the Cartesian plane under the identifications

  
    
      
        (
        x
        ,
        y
        )
        ∼
        (
        x
        +
        1
        ,
        y
        )
        ∼
        (
        x
        ,
        y
        +
        1
        )
        ,
        
      
    
    {\displaystyle (x,y)\sim (x+1,y)\sim (x,y+1),\,}
  or, equivalently, as the quotient of the unit square by pasting the opposite edges together, described as a fundamental polygon ABA−1B−1.
Based on the most recent criteria, classification with APS requires one clinical and one laboratory manifestation:
Clinical:A documented episode of arterial, venous, or small vessel thrombosis — other than superficial venous thrombosis — in any tissue or organ by objective validated criteria with no significant evidence of inflammation in the vessel wall
1 or more unexplained deaths of a morphologically normal fetus (documented by ultrasound or direct examination of the fetus) at or beyond the 10th week of gestation and/or 3 or more unexplained consecutive spontaneous abortions before the 10th week of gestation, with maternal anatomic or hormonal abnormalities and paternal and maternal chromosomal causes excluded or at least 1 premature birth of a morphologically normal neonate before the 34th week of gestation due to eclampsia or severe pre-eclampsia according to standard definitions, or recognized features of placental insufficiency
Laboratory:Anti-cardiolipin IgG and/or IgM measured by standardized, non-cofactor dependent ELISA on 2 or more occasions, not less than 12 weeks apart; medium or high titre (i.e., > 40 GPL or MPL, or > the 99th percentile)
Anti-β2 glycoprotein I IgG and/or IgM measured by standardized ELISA on 2 or more occasions, not less than 12 weeks apart; medium or high titre (> the 99th percentile)
Lupus anticoagulant detected on 2 occasions not less than 12 weeks apart according to the guidelines of the International Society of Thrombosis and Hemostasis.There are 3 distinct APS disease entities: primary (the absence of any comorbidity), secondary (when there is a pre-existing autoimmune condition, most frequently systemic lupus erythematosus, SLE), and catastrophic (when there is simultaneous multi-organ failure with small vessel occlusion).According to a 2006 consensus statement, it is advisable to classify APS into one of the following categories for research purposes:

I: more than one laboratory criterion present in any combination;
IIa: lupus anticoagulant present alone
IIb: anti-cardiolipin IgG and/or IgM present alone in medium or high titers
IIc: anti-β2 glycoprotein I IgG and/or IgM present alone in a titer greater than 99th percentileThe International Consensus Statement is commonly used for Catastrophic APS diagnosis.