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nL-n-Cv1: Morley’s Mono EnnaCardioid

nL-n-iCv1: Morley’s Multiple EnnaCardioids

nL-n-Cv1 and nL-n-iCv1 are curves described by Morley in Ref-37 and Ref-47, with the notation Cn.

The name EnnaCardioid is a generic name introduced by Morley in Ref-47, On Reflexive Geometry, page 15. The names Cardioid, TetraCardioid and PentaCardioid are also used by him.

  • In a 3-Line it is a circle (Morley Code C2).
  • In a 4-Line it is a regular Cardioid (Morley Code C3).
  • In a 5-line it is called a TetraCardioid (Morley code C4).
  • In a 6-line it is called a PentaCardioid (Morley code C5).
  • etc.

Morley’s pupil Edward C. Philips wrote a dissertation (Ref-56) “On the PentaCardioid” in 1908. In this paper he gives a general description of the different types of PentaCardioids that occur as well as a method of constructing the PentaCardioid.

Each nL-Mono EnnaCardioid nL-n-Cv1 is circumscribing n lower-level (n-1)L-EnnaCardioids.

All nL-Multiple EnnaCardioids nL-n-iCv1 are inscribed in the corresponding n-Line and their centers lie on Morley’s Axes nL-n-iL1.

When n=4 it is a regular Cardioid. At higher levels it is a special type of curve often with many cusps.

See also QFG messages #815-#825, #831.

Infovisual nL-n-Cv1-infovisual-cvt-01.png
Mono EnnaCardioids and Multiple EnnaCardioids
  • In a 3-Line there are:
    • 1 Mono Circle C2 here coded 3L-n-Cv1 (circumcircle),
    • 4 (=22) Multiple Circles C2 here coded 3L-n-4Cv1 (in-/excircles) touching the defining 3 lines of the 3-Line.
  • In a 4-Line there are:
    • 1 Mono Cardioid C3 here coded 4L-n-Cv1 (in EQF QL-Qu1) circumscribing the Mono Circles of the 4 Component 3-Lines of the 4-Line,
    • 33 Multiple Cardioids C3 here coded 4L-n-27Cv1 (in EQF QL-27Qu1) touching the defining 4 lines of the 4-Line.
  • In a 5-Line there are:
    • 1 Mono TetraCardioid C4 here coded 5L-n-Cv1 circumscribing the Mono Cardioids C3 of the 5 Component 4-Lines of the 5-Line,
    • 44 Multiple TetraCardioids C4 here coded 5L-n-64Cv1 touching the defining 5 lines of the 5-Line.
  • In a 6-Line there are:
    • 1 Mono PentaCardioid C5 here coded 6L-n-Cv1 circumscribing the Mono TetraCardioids C4 of the 6 Component 5-Lines of the 6-Line,
    • 55 Multiple PentaCardioids C5 here coded 6L-n-3125Cv1 touching the defining 6 lines of the 6-Line.
  • etc.
Construction
  • For a construction of Morley’s Mono Cardiod in a 4-Line see EQF, QL-Qu1.
  • For a construction of Morley’s Multiple Cardiods in a 4-Line see EQF, QL-27Qu1.
Construction of 5L-n-Cv1

(example in a 5-Line by Eckart Schmidt, see QFG#815)

  1. Let Q be a variable point on the 5L-n-Ci1-circle,
  2. let Ci1 be a circle round Q through 5L-n-P1,
  3. let X be the second intersection of Ci1 and the QL-P1-circle,
  4. let Y be the second intersection of X.5L-o-P2 and Ci1,
  5. let Ci2 be a circle round Y through X,
  6. let Z be the second intersection of Ci1 and Ci2,
  7. then Z reflected in Y is a point P of Morley’s EnnaCardioid.
Correspondence with ETC/EQF
  • When n=3, then nL-n-Cv1 = circumcircle of the 3-Line.
  • When n=3, then nL-n-iCv1 = combination of incircle and excircles of the 3-Line.
  • When n=4, then nL-n-Cv1 = QL-Qu1.
  • When n=4, then nL-n-iCv1 = QL-27Qu1.


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