Rees Hawkins Acheson contacted me the other day and I wanted his story to be told about the development of the Hawkindale Timber Framing Angles and credit for him developing the timber framing angles on his own in 1987.

Mr. Ayers,

I tried to post the following on your blog:

but was unable to do so.
I obtained your email and have included post

below:

I appreciate your short history of the framing angles and
I am pleased that someone is taking it on.
But I am also writing to point out that you got my name wrong, the name
is Rees Acheson.

While I'm at it, you may find interesting the origin of
the name "Hawkindale". While
working at Benson Woodworking (1987-1991) I developed a set of angles to
complement the Martindale angles (I had not known of Mckibben). My recollection is that all my angles would
have been developed between November, 1987 and February or March, 1988. Tedd Benson suggested that these new angles
be named after me as Acheson Angles. I
disagreed and wanted to keep Martindale's name.
So we made a compromise and choose Hawkindale - a combination of my
middle name, Hawkins, and Martindale.
This seemed to fit well with a concatenation of the angles, making one
list.

Further, it might be interesting to note that your
description seems to list Ed Levin and Rees Anderson as co-developers in the
angle development process. This is
misleading in that I talked to Mr. Levin once in the late spring or summer of
1988 (I remember it was green

outdoors) when Tedd brought the two of us together, and
perhaps again in

1990 after the Timber Framer's Guild conference where I
gave a talk. I do not remember the
second meeting but the first was brief, about 5 minutes, in which I told Mr.
Levin that in my coding I was describing the joints exclusively with
angles. Having said that, apparently the
conversation was over and, other than niceties, no more was said - I learned
nothing of what he was up to. My guess
is that he either thought my approach was brilliant or ludicrous. He did not say. My code comments below indicate a 2nd
meeting, or at least that a communication, took place.

We were independent, and although I made every effort to
share what I had done with anyone who asked, Mr. Levin and I did not
collaborate.

Your blog does not suggest that most of the
non-Martindale angles were mine. Perhaps
you could contact Tedd Benson for clarification on that matter. tedd My involvement in compound joinery was brief,
about three years, after which I went back to my profession in machine design
which I feel better suits my strengths.
I am unfamiliar with what occurred with compound joinery after I left,
but I am quite familiar with my doings for those three years.

What follows is the computer code for calculating the
Hawkindale angles from my HV computer program that I began writing in late 1987
for Benson Woodworking. The file date is 27 October, 1990, so the final
modification was apparently of Mr. Levin's influence (see comment for P7). The comments, in curly brackets {}, indicate
that angles A1 thru

A9 were mine and that early on I changed their names to
fit in with the Martindale scheme (e.g. A1 became P4). Two angles appear to have had Ed Levin's
involvement in 1990 (P7 & R6) and the rest are Martindale's.

Mr. Levin's R6 seems to have replaced my R6 for some
unstated reason.

Due to dependencies, they are not necessarily in order.

(I think Martindale's are as you say: R1, R2, R3, P1, P2,
P3, C1-C9).

R1 :=
ATan(Tand(S) * Sind(D));

R2 :=
ATan(Sind(S) * Cosd(S) * Cosd(D) / Tand(D));

R3 :=
ATan(Sind(S) * Cosd(S) * Cosd(D) / Sind(R1));

P1 :=
ATan(Sind(S) / Tand(D));

P2 :=
ATan(Cosd(S) / Tand(D));

P3 :=
ATan(Cosd(D) * Sind(R1) * Cosd(R1) / Cosd(S));

C1 :=
ATan(Sind(P1) / Tand(S));

C2 :=
ATan(Tand(R2) * Cosd(R3));

C3 :=
ATan(Cosd(S) / Tand(D));

C4 :=
ATan(Sind(R1) / Tand(D));

C5 := C4;

C6 :=
ATan(Tand(D) / Sind(R1));

P4 := P2 -
Atan(Tand(R2) * Tand(C5) * Cosd(C5));

{Formerly A1}

P5 := ATan
(Sind(D) *(Cosd(S) /Cosd(D)) ); {5/6/88
}

{formerly A2}

P6 := Atan
(Tand(C5) * Cosd(90-p2)); {1/22/89} {c5 rotated by 90-p2}

{P6
formerly A3} {OK
1/12/90} {JP seat cut}

P7 := P1 -
(ArcCos(Cosd(R3) / Cosd(C1)) ); {Ed
Levin 10/26/90}

R4 := Atan
(Tand(p2) /Cosd(c5) );

{Formerly A4}

R5 := ATan ( Tand(r1) * Cosd(D) );

{Formerly A5}

{r1 rotated by D}

{R6 := Atan
(Tand (P6) * Cosd (P2) );}
{Obsolete}

{Formerly A8}

R6 := Atan
(Tand(90-R5) / Cosd(D)) - (90-R1); {Ed
Levin 10/26/90}

{ R6 =
(the angle from the edge of rafter to the edge of bottom }

{ housing of VR foot

[p6]). }

{ It exists on as the seat cut on bottom of
VR Ft }

A7 := ATan (
Tand(S) * Cosd(D) ); { S rotated by D or
C5 rotated by r1.}

{ Same
as c5 but a plum section of VR}

{
Angle of Roofs in a plum

section }

{
normal to the Valley

Line. }

A8 := Atan
(Tand(r2) * Tand(c5) * Cosd(c5));

A9 := Atan
(Tand(R1) * Tand(C5) * Cosd(C5));

February 2, 2014

Rees Hawkins Acheson

Sim,

To answer your questions:

"Does HV programming refer to Cobol computer
programming?"

All programming
was done in pascal, a compiled language that was very fast, both to compile and
to execute. It's too bad the language
never really caught on.

"HV"
(stands for "Hip & Valley") was a computer program I wrote to
create drawing files for AutoCAD to read.
These files contained all the necessary views for the shop to cut the
member. A valley Rafter Peak Tenon might
have 11 views. The program would take
about 5 seconds to run and create the drawing - with more than four of those
seconds doing the relatively slow disk output.
This was in 1987 when the CPU speeds were only 40MHz or so. The program was very fast. If it could be done at all, doing the same
task in AutoCAD's AutoLisp scripting language would probably have taken an hour
or two to run.

"1: How did you prove or test your formula's in
1980's? Especially the formula for angle P4."

To find the
angles I worked with a pencil on a drafting table. Angles were worked out on E size sheets of
paper. One angle lead to the next along
with its formula and derivation for each stage written there on the paper until
the desired angle was obtained. As
proof, once a desired angle was worked out, I would work it out from another
perspective - using different givens or different proven angles - coming at the
angle from another direction. The
results of the two must match. To
clarify: I would find at least a 2nd formula for the same angle (sometimes a
3rd or 4th) and their results must match.

I think there
were finally seven or eight such work sheets, each completely filled up with
calculations, notes and drawings. I do
not know what eventually became of these sheets, they may still exist somewhere
and if so, I would like to see them again.
These sheets were very valuable to me.

I do not remember
P4 as being a particularly difficult one.
Each one of them brought their own struggle in which I would grapple
with with their complexities until I figured it out. And I did struggle.

I only remember
one angle as being real a problem, A2, and it makes for an interesting story
(A2 later became P5). In fact, I could
not get it - the formula would not work, always pretty close but no cigar.

Looking at my notes it seems that the Y and Z components
of a test point were correct but X was always off by a small amount (in what
plane I do not now know). This should
have been a clue, but I did not yet understand how. Finally, I had spent so much time on this
angle that I needed to move on. I wrote
an iterative solution (computer) that worked out the angle buy narrowing in on
it using two methods of calculating the X component (working at it from two
different sides) until the error was small enough (>0.00001") and then
calculating the angle from the now correct X, Y & Z. This worked well enough that I found no need
to fix it (it ran the hundred or so iterations necessary in about 200ms, so
waiting for it was not an issue). But in
early 1990, two years later, Tedd Benson asked that I give a talk on what I had
been up to for that summer's TFG conference.
This terrified me. But it also
meant that I needed to figure out A2.

I began again
using my original sheets on the drafting board.
It's difficult to believe now, but I spent several days working on this
one angle. Finally one day I stopped,
disgusted, and decided to go for a walk to clear my mind. On that walk I realized something and when I
got back, sure enough, I had an incorrect initial assumption. It was so near the beginning, it had been a
given, that I had overlooked it each time through. Fixing this made the formula work fine. This became one of those learning experiences
that one never forgets. I cannot
remember exactly what the bad assumption was but it was pretty basic - something
like the starting angle not being in the plane of the paper but very close to
it. I knew better than that and was
amazed that I had not noticed.

"2: Did you draw out roof framing kernels? Like the
French?"

Sorry, don't know
what this is.

"3: Did you use tetrahedrons in the 1980's?"

No, I don't think
so. But I'm not sure what the method is,
either.

Looking up tetrahedrons on your blog, a cursory look
suggests the use of geometry to unfold views.

As I recall, most
of what I was doing, was after seeing that an angle was needed, I would begin
rotating other known angles until I got to the desired plane of
measurement. This was in my head at
first, then once I found a way I would begin the real rotations on paper. Obviously, after thinking through several
rotations I would be in danger of becoming "lost", necessitating
starting again. But this was less
restricting than the interference created by having to leave one's mind to
write things down. The series of
rotations would be the formula (e.g. to pick a few angles at random: R1 by A7
by C1, etc.). Of course, I also needed

to keep track of rotation axes and the like. There might be a great

many rotations involved and the formula long. Once found, though, the formula would be
simplified. It was often impossible to
deduct the roots of the formula once simplified. Those original sheets of paper
were therefore very valuable to me. They
told the story and I would use them later to prove other things. So, as to your curiosity regarding R4, I do
not remember - it's on those sheets.
Right now I can't even remember what R4 is used for.

As I mentioned
previously, as a proof, I would work out at least one other way to get to the
same angle (producing a different formula).
It was always best to use few angles that were common to the formulae
being tested. It was also best to obtain
the angle from a completely different starting place. The more independence the higher the degree
of confidence I could have. If each
component was unique, just two formulae would suffice.

To check that
these formulae were similar I had written a computer test jig. In it the list of formulae to be compared
would be run with a series of data to test a spectrum of possible input
comparing the output. The output of each
formulae would have to agree within a rather small "CloseNuf"
constant. At the time I was using 80 bit
floating point numbers (called "Extended" in pascal) and so the
wandering due to processor error was very small. Errors were probably mine.

To help with
angle development I wrote a short computer program that would rotate numerical
angles. In use, after entering test roof
pitches and dormer angle it would calculate the Hawkindales. Then it was ready to do test rotations. I had worked out a simple language for data
entry and it would spit back a resultant numerical angle. This result could be further rotated again
and again. The result could be checked
for reasonableness and became a valuable tool for me to see if I was on the
right track. If desired, data entry
could use the Hawkindale angles for rotation.
For example a rotation instruction might be "r4,a" which would
rotate the currently displayed angle by Hawkindale R4 around the "a"
side of the angle as the axis of rotation.
Entries such as (90-R4)+C1,a were also possible. This was very much quicker than using a
manual calculator and entering the numbers.

Looking through my notes I see that the change in
"A" angle names occurred on Jan 16, 1990. Presumably this was because of the upcoming
TFG talk I was to give that summer.

I hope that this adequately answers your questions. However, I fear that my response to your R4
query has been disappointing. But
somehow I also suspect that I may have been going about angle development differently
than most people and so how I arrived at R4 might not answer your question
anyway. I have no formal training and so
I developed the methods myself.

Rees

I would like to try answering question 3 once more. What I said was not clear and while thinking
it over some more I have remembered more of it.
So I will try again:

"3: Did you use tetrahedrons in the 1980's?"

No, I don't think
so. But I'm not sure what the method is,
either.

Looking up tetrahedrons on your blog, a cursory look
suggests the use of geometry to unfold views.
Although this may have been what I was actually doing, it was not in
such a formalized fashion.

As I recall, upon
realizing the need for a new angle, I would begin rotating planes or other
angles until they were in the plane of the angle I wanted. Sometimes this would mean starting with the
two lines of the angle I desired and perform rotations until it was in a plane
I could deal with. Other times I would
start with a plane I was comfortable in and perform rotations until I was in
the plane of the angle I wanted. Either
way, I would end up with an angle calculation in its own or some other plane
followed by a series of rotations along various axies. The rotations were usually by a Hawkindale
angle, but, of course, all angles had to be in terms of the original angle
input - the roof pitches (S angles) and dormer angles (D angles).

This would be
done in my head first, then once I found a probable solution I would begin the
real rotations on paper. Obviously,
after thinking through several rotations I would be in danger of becoming
"lost", necessitating starting again.
But this was less restricting than the disturbance created by having to
leave one's mind to write things down.

There might be a
great many rotations involved and the formula long.

Once found, though, the formula would be simplified. It was often impossible to deduct the roots
of the formula once simplified. Those original sheets of paper were therefore
very valuable to me. They told the story
and I would use them later to prove other things. So, as to your curiosity regarding R4, I do
not remember - it's on those sheets.

As I mentioned
previously, as a proof, I would work out at least one other way to get to the
same angle (producing a different formula).
It was always best to use few angles that were common to the formulae
being tested. It was also best to obtain
the angle from a completely different starting place. The more independence the higher the degree
of confidence I could have. If each
component was unique, just two formulae would suffice.

To check that
these formulae were similar I had written a computer test jig. In it the list of formulae to be compared
would be run with a series of data to test a spectrum of possible input
comparing the output. The output of each
formulae would have to agree within a rather small "CloseNuf"
constant. At the time I was using 80 bit
floating point numbers (called "Extended" in pascal) and so the
wandering due to processor error was very small. Errors were probably mine.

Of course, the
final proof was when the HV program used these formulae to create the 3D points
of a joint. If a formula were incorrect
at least one of the ordinates of a point would be measurably and probably very
noticeably incorrect.

To help with angle
development I wrote a short computer program that would rotate numerical
angles. In use, after entering test roof
pitches and dormer angle it would calculate the Hawkindales. Then it was ready to do test rotations. I had worked out a simple language for data
entry and it would spit back a resultant numerical angle. This result could be further rotated again
and again. The result could be checked
for reasonableness and became a valuable tool for me to see if I was on the
right track. If desired, data entry
could use the Hawkindale angles for rotation.
For example a rotation instruction might be "r4,a" which would
rotate the currently displayed angle by Hawkindale R4 around the "a"
side of the angle as the axis of rotation.
Entries such as (90-R4)+C1,a were also possible. This was very much quicker than using a
manual calculator and entering the numbers.

Rees

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