# [Origami] Summary: Terminology question

```Den 8. jul. 2019 kl. 12.11 skrev Papirfoldning.dk <h...@papirfoldning.dk>:
> When I fold side to side I make a book fold and I have created a ... what?
> When instructing I can tell them to fold sides to sides, or to make the ...
> what?  ```
```
> When making each of these steps, I get the diagonal cross and the ... what
> ... cross?

Thank you for all the answers. I think you have seen them in the list.
My personal favourite is median, with bimedian, middle line, and orthogonal as
runners up.

Below I summarise the suggestions and add some extra analysis.

The suggestions are all good, and I am pretty sure that all of them will get
the meaning across in a teaching situations. Still, there are some anomalies,
and when writing about folding, I like to be able to refer to the folds more
rigorously.

Some suggested just to fold along, e.g.:  "fold sides to sides" or "fold the
paper in half Side-to-Side (this manoeuvre is usually called a Book Fold), then
Open, Rotate, and Repeat", without actually referring to the lines created. The
focus here is to learn how to make the fold, rather than the name of it.

I often do that as well: just instruct what to do, without naming the
intermediate results. However, in some cases I prefer stating the subgoal of a
series of folds before doing them, e.g. when folding a jumping frog: "We will
make a diagonal cross here (points). To do this ... (showing and explaining
techniques as needed)...". That provides the students with a better
understanding of why we do the folds.

One of the above responders also mentioned "fold the paper in half Diagonally,
then Open, Rotate, and Repeat". Interestingly, if this should be symmetrical to
the "side-to-side", the instruction should have been "fold the paper in half
point-to-point (this manoeuvre is usually called a Triangle Fold), then Open,
Rotate, and Repeat", without actually referring to the lines created.

This situation occurs because the suggester has a name, "diagonals", in one
case and not in the other case, thus being forced to speak about it without
referring to the object created. Precisely the predicament that led to my
original question above.

Two answers suggested using visual analogies: "diagonal folds make an 'X' and
book folds make a '+'  (cross)". A nice, informal solution. I don't get why '+'
is a cross and 'X' is not called that, but maybe that is just an omission.

Some responders gave names they use. Interestingly, no two responders suggested
the same name.

One responder suggested to call them the horizontal and vertical lines. Even
more precise than what I asked for; diagonals don't have that degree of
precision.

Another suggestion is "orthogonal" as the lines are at 90 degrees with the
sides. It is pretty analogous to what I call it in Danish, "Tværfolder". To
walk "på tværs" of a place is to cross it from side to side, as opposed to
crossing it diagonally.

Side note: A Danish origami book from 1934 also calls those folds "tværfolder".
The book is "Lærebog i Dansk Skolesløjd" (Textbook of Danish School Crafts)
published by the Danish Craft Teacher Association. 150 pages of the 331 total
are origami related. The book has references to Friedrich Fröbel, but also
cites Pestalozzi, Salzmann, Rousseau, John Locke, and Amos Comenius.

A third suggestion is "middle lines". Informally easy to understand, and places
the emphasis on where the line is placed, similarly to the angular "bisector"
which in a square is the same as the diagonal.

A fourth suggestion is "median", with the argument that everything as a
geometrical name. I really like this as it is a term that mirrors "diagonal" at
the stylistic level, seems well defined and matches the colloquial uses. The
only problem is that in English (and Danish) a median mathematically is tightly
bound to triangles. In English the mathematical term seems to be  "bimedian"
(http://mathworld.wolfram.com/Bimedian.html). A somewhat pleonastic term, just
"median" would have been the logical choice.

Then we could call it bimedian, except that it is not a commonly known term. At
least Merriam-Webster does not know it:
https://www.merriam-webster.com/dictionary/bimedian. More surprisingly,
Encyclopedia of Mathematics doesn't know it, either
(https://www.encyclopediaofmath.org/index.php?title=Special%3ASearch&search=Bimedian&button=).
For what it's worth, the spell checker on iPad does not know "bimedian"
either, and places red dots under that word.

The responder to suggest median is French, and indeed in French median is the
term we look for: https://fr.m.wikipedia.org/wiki/Médiane_(géométrie).

Let us investigate median a bit further.

In general a median is the middle of something and divides that something into
equal parts or goes mid between two sides, paraphrasing Merriam-Webster
(https://www.merriam-webster.com/dictionary/median). This fits well with what
we need it to mean.

In analogy with this colloquial formulation, and with the French definition,
this site https://rechneronline.de/pi/square-calculator.php calls the lines
connecting the opposite midpoints for median lines.
Incidentally, RechnerOnline.de is a German site, though a number of pages are
in English.
Most hits on “median lines” is a theory of market price behaviour:
https://www.esignal.com/publicdocs/eSignal_Manual_ch18.pdf.

A median in geometry is normally a property of triangles: The line that goes
from a corner to the opposite edge. At Wikipedia there is no indication
whatsoever of other interpretations
(https://en.m.wikipedia.org/wiki/Median_(geometry)), except that it extends to
tetrahedra which essentially are 3D-triangles.

For trapezium a median (https://www.mathopenref.com/trapezoidmedian.html), also
called a midline, is a line connecting the midpoints of the two legs of a
trapezoid. Incidentally, for the special case where the legs and the parallel
sides are the same length and orthogonal, the trapezoid is a square, and the
median connects the midpoints of opposite sides of the square.
Meriam-Webster knows this definition of median, the above reference has it as
sense 3.b, again indicating that median is the more well-known concept.

For triangles, though, medians and midlines are not the same. A triangle
midline is a line connecting the midpoints of two sides. So the generalisation
of midlines in a square would be any lines connecting midpoints of any two

In 1962 Roger B Kirchner wrote "A Generalization of the Median Theorem for
Triangles," The American Mathematical Monthly, 69(7), 1962 p. 650 and in 2011
he contributed a demonstrator to Wolfram Alpha:
http://demonstrations.wolfram.com/MedianTheoremForPolygons/.  He defines
medians inductively: The median of an 1-gon is the point itself, a 2-gon median
is the midpoint of the line between the two points, and for a n-gon the median
is a line from the midpoint of a k-gon formed by k of the points to the
midpoint of the n-k other points, 0 < k < n.

In this generalisation a square has two kinds of medians. 0 < k < 4, but k=1
and k=3 ends up symmetrical. With k=1, the median goes from a point to the
midpoint of the triangle formed by the other three points. With k=2, the median
goes from the midpoint of a side to the midpoint of the opposite side.

The latter corresponds to the trapezoid median whereas the case k=1 is a bit
surprising but turns out to be a meaningful abstraction. With this definition,
the generalised medians preserve the central median theorem: All medians
intersect in a single point which the midpoint of the n-gon. This midpoint is
also known as the centroid, or the gravitational midpoint of the polygon, and
may be computed as the mean of the points of the polygon.

Enough said. Go fold some diagonals and medians ...

Best regards,
Hans

Hans Dybkjær
papirfoldning.dk
Society: foldning.dk```