arithmetic – The Pleasure of Hexasect


This geometric development problem is
a set of parts to be positioned so as,
def s #1{{ smallsf #1 }}
def AB { overline {s{AB}} }
def line #1{{ small overleftrightarrow {s{#1}} }}

dividing line phase $AB$ of size 6
into 6 subsegments, every of size 1.

hexasect – transitive
verb – to divide into 6 equal elements

How can $AB$ be hexasected by inserting 5 circles and seven traces
that produce simply 1 lines-only node?

Development tips

Circles and contours are positioned sequentially,
in any order that accords with nodes that exist at instances of placement.

A circle could also be positioned the place a node exists for the circle’s heart.

A line could also be positioned the place it crosses a minimum of two current nodes.

Nodes are endpoints A and B in addition to
intersections amongst circles, traces and/or $AB$.

A lines-only node is an intersection of traces within the accomplished resolution.
No circles go by means of a lines-only node.
(Nodes alongside $AB$ are ineligible as a result of $AB$
is technically a phase, not a line.)


quadrisect – transitive
verb – to divide into 4 equal elements

 A special $AB$, of size 4, might be quadrisected
by inserting 4 circles and 6 traces
that produce simply 1 lines-only node.

• Step 1
locations circles centered at nodes A and B.
These circles’ intersections produce nodes C and D.

• Step 2
locations line $line{BC}$ and a circle centered at C,
whose intersection produces node E,
and locations $line{CD}$,
whose intersection with $AB$ produces node F.

• Step 3
locations a circle centered at E.
This circle’s intersection with $line{BC}$
produces node G.

• Step 4
locations traces $line{AD}$ and $line{FG}$,
whose intersection produces node H.

• Step 5
completes this quadrisection
by inserting traces $line{CH}$ and $line{DG}$.
The specified lines-only node is H.