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.
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.
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.
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 5L-o-Ci1-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.
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.