Re: The TRIRADIUS in a fingerprint: how it develops, it's characteristics + a definition!

**Lynn** Tue Apr 26, 2011 11:05 am

Thanks for taking the trouble to find my quote Patti.
I think I should remove the red words at the end, as that's what Martijn picked up on and replied:

your conclusion is 'firm', but these words sound like you misunderstood my point, because you deny something... that I didn't state at all:

"A bifurcation is not the same thing as a triradius."

Again, I didn't claim that 'bifurcations' and 'triradii' are the same - because of the angle difference! But they do belong to the same FAMILY of ridge constellations:

After all the discussions, I think Patti and I are now both quite clear in our minds about the various manifestations of triradii. We both prefer to differentiate between a bifurcation and a triradius.

**Martijn (admin)** Tue Apr 26, 2011 1:29 pm

Lynn wrote:Thanks for taking the trouble to find my quote Patti.
I think I should remove the red words at the end, as that's what Martijn picked up on and replied:

your conclusion is 'firm', but these words sound like you misunderstood my point, because you deny something... that I didn't state at all:

"A bifurcation is not the same thing as a triradius."

Again, I didn't claim that 'bifurcations' and 'triradii' are the same - because of the angle difference! But they do belong to the same FAMILY of ridge constellations:

After all the discussions, I think Patti and I are now both quite clear in our minds about the various manifestations of triradii. We both prefer to differentiate between a bifurcation and a triradius.

Hi Lynn,

Thanks for making that correction regarding your use of the red colour.

But you just made another statement that isn't clear (nor specified) at all:

"We both prefer to differentiate between a bifurcation and a triradius."

- First of all, this comment suggests (again) something that I did not state: because I do recognize that in the 'scientific approach' the angles between the diverging ridges are a decisive factor that discriminates a bifurcation from a triradius. (However, I also observe that in the F.B.I. system this discrimination is no longer made... just like the fact that the use of the word 'triradius' became completely irrelevant!!!).

- Second, in your orange comment above you haven't specified anything about the word 'bifurcation'. And because the missing of specification the comment becomes a little bit confusing - because there are 2 perspectives where the word 'bifurcation' is used in a different manner. In the F.B.I. system the words is used in a much broader way (and Patti actually has explicitely agreed [= agreement 1] with me about that, she proposed a specification and I confirmed that specification). And contrary to the 'scientific approach', in the F.B.I. system any triradial point that looks similiar to a 'scientific triradius'... can be described as a 'bifurcation': again, see the example of figure 139 that I mentioned yesterday, where Patti did not disagree about that example as well [= agreement 2].

- Third, I think my essential point is that in the F.B.I. system there is no alternative 'label' that could serve as a synonym for a 'triradius' - only the word bifurcation can serve for that role. And Patti does not appear to deny that either [= agreement 3]... but still she states:

"I do not agree that the word bifurcation is an exchangeable term for triradius."

I think this statement of Patti is actually true for one of the two perspectives: the 'scientific system', because of the specification regarding the angles.

But the conflicting element between this statement of Patti and her earlier agreements [agreement 1 to 3], is that the word bifurcation is used in a broader manner in the F.B.I. system... without any specification at all regarding the angles between the diverging ridges.

And therefore I have one question for you to answer:

Lynn, does your statement also implicate that you can agree with Patti conclusion that the F.B.I.'s use of the word 'bifurcation' is: ... (I quote from Patti's words:) ... "misleading" & "misusing"?

PS. I hope you remember my PM's where I described how the word 'triradius' became an 'outdated' word during the past 25 years since the first publication of the F.B.I. book in 1985 - my Google Scholar statistics clearly demonstated how in time the word 'delta' became much more often used that the word 'triradius'.

**Martijn (admin)** Tue Apr 26, 2011 1:43 pm

Patti wrote:
Martijn wrote:
Patti... ; sounds to me that you made at least some progress in recognizing that the F.B.I. really has only one single 'label' available that could be used for describing a 'triradius': the word 'bifurcation'.

I do not agree that the word bifurcation is an exchangeable term for triradius.

I think it is a mistake to use "bifurcation" when describing a triradius.

Hi Patti,

I understand what you are saying, but then... do you see any other fingerprint-element in the F.B.I. system that could be used to describe a 'triradius'?

(The word 'delta' can not serve for that purpose. Because - despite the example of the triangle in the river-delta - regarding fingerprints the F.B.I.'s word 'delta' can only serve as a synonym for the word 'triradial point'... and not as a synonym for the much larger shape of a complete 'triradius')

PS. By the way, I am aware that Coppock definition for a triradius is of course presented from the scientific perspective. But it does show EXPLICITELY that even in the scientific perspective the concept of a 'triradius' is associated with the concept of a 'bifurcation'... only the angle show the difference. And therefore the F.B.I.'s more wider use of the word 'bifurcation' does make sense... and is not 'misleading' anyhow at all!

**Lynn** Tue Apr 26, 2011 3:00 pm

Second, in your orange comment above you haven't specified anything about the word 'bifurcation'. And because the missing of specification the comment becomes a little bit confusing - because there are 2 perspectives where the word 'bifurcation' is used in a different manner.

to quote FBI - A bifurcation is the forking or dividing of one line into two or more branches.
what other specification is there?

Patti Tue Apr 26, 2011 3:47 pm

A bifurcation is 1 ridge, from 1 ridge field or system, that forks or splits and continues on as two ridges. One branch continues in the same direction and field system as the mother ridge and the other ridge typically flows in the same parallel direction. A bifurcation's forks are usually 45 degrees or less apart.

A triradius is 3 separate ridges that are from 3 different ridge fields or systems that meet at the center or at the outer angles. The angles between the 3 ridges is 'near' equal ranging from 90 to 120 degrees.

The FBI does not use the same terminology to locate the delta/triradial point.

FBI delta and Triradial points are only 'imaginary' focal points. They are the same location regardless of the system used.

The FBI illustrates the Triradius without calling it the Triradius in their image of the 2 fields and 1 lake. 3 Fields meeting. The triangular shaped 'delta' is located where the 3 fields meet. A side of that triangle in their sketch is facing the pattern rather than a pointed side. It's open to the core.

The FBI is clear in the description of a bifurcation, as Lynn described. A single ridge forking. Yet, they use the word bifurcation to represent a forking or V shaped pattern of ridges in the area that we come upon via 'innermost parallel diverging ridges'.

There is a second 'use' for 'bifurcation' by Cummins & Midlo along with the FBI. Both refer to what is found inside the diverging parallel ridges (FBI looks at 1 set, Schaumann & Alter mention 3 sets of parallel diverging ridges) when a V shape they call it a *bifurcation*.

In doing so they ignore all the rules for a bifurcation. I think they are assuming they are just saying the V shape commonly found looks like a splitting ridge and most will not concern themselves with the misnomer.

This is all that I am agreeing to. I do not think a bifurcation qualifies as a triradius. It is but 1 ridge from one field becoming 2, and a triradius is always 3 separate ridges from 3 different fields.

**Lynn** Tue Apr 26, 2011 4:12 pm

Patti, where did you get this from?

A bifurcation's forks are usually 45 degrees or less apart.

I'm sure that's not correct.

Patti Tue Apr 26, 2011 4:33 pm

Lynn wrote:Patti, where did you get this from?
A bifurcation's forks are usually 45 degrees or less apart.
I'm sure that's not correct.

Yesterday I quoted from FBI's info on Tented Arches.

A tented arch is different from a plain arch in that it has a triradius.

In the rules they tell us that if an upthrust from a ridge is less than 45 degrees it does not count. By deduction, that would mean the ridge rising from another ridge, is a fork or a bifurcation. Could be an appendage of some kind, dot or island? I'd see it as a short rising branch between two parallel ridges.

That ridge if at least at a 45 degree angle divides the single field into distal and proximal fields.

If that upthrust is at a 90 degree angle to the other ridges (and the other ridges are fairly horizontal in an arch) 3 fields are divided. Proximal and the ulnar and radial lateral sides.

This is how I interpreted the FBI's instructions.

**Martijn (admin)** Tue Apr 26, 2011 4:54 pm

Lynn wrote:
Second, in your orange comment above you haven't specified anything about the word 'bifurcation'. And because the missing of specification the comment becomes a little bit confusing - because there are 2 perspectives where the word 'bifurcation' is used in a different manner.

to quote FBI - A bifurcation is the forking or dividing of one line into two or more branches.
what other specification is there?

Thanks Lynn,

Yes, your specification confirms to me that you had 2 perspectives in mind (because the word 'triradius' relates only to the scientific perspective; while a 'bifurcation' is defined different in both perspectives).

Okay, if we focuss on the details in the F.B.I. definition for a 'bifurcation' then I might be able to explain my point more explicitely:

Because I think this F.B.I. definition clearly describes that a 'bifurcation' can manifest in various shapes: (1) "the forking", and (2) 'or dividing' of one line into two branches.

And I also think it is obvious that only the second aspect directs us to the 'bridge' described by Coppock, who's definition describes that there is no fundamental difference between a 'bifurcation' and the typical manifestion shape of a 'triradius'... only the issue of the angle provides a tool to describe the difference.

Indirectly this is also confirmed by Loesch who mentioned that in practice 'triradii' typically manifest as an asymmetrical shape - which is also typical for any 'bifurcation' (due to the round, bulbing surface of a fingertip, etc).

So, while the scientific approach used to discriminate:

1 - a 'splitting', as a (asymmetric) divergence of one ridge into 2 ridges that follow same direction (no angle);
2 - a 'fork'/'Y-shape', as a (symmetric) divergence of one ridge into 2 ridges that follow the same direction (no angle);
3 - a 'bifurcation', as a divergence of one ridge into 2 ridges in different directions (angle smaller than 90 degrees);
4 - and a 'triradius' as a divergence of one ridge into 2 ridges in different directions (angle larger than 90 degrees up to 120 degrees).

... times have changed, and the F.B.I. introduced a much more simple approach:

The F.B.I. introcuded a multi-concept definition for a 'bifurcation' - excluding any specifications regarding: (a) the angle or (b) the issue of whether ridges continue as parallel ridges or in a different direction.

And as a result of that it became obvious for me that the F.B.I.'s alternative definition for a 'bifurcation' ... can perfectly serve as a substitute for:the 'splitting', 'fork' / 'Y-shape', and the 'triradius'!

And therefore I say that there is no need at all to declare the F.B.I. approach as 'misleading' (nor 'misusing' regarding how they re-define the word bifurction).

Because in a way, ... Smile

... the F.B.I. has made fingerprint interpretation 'a lot more simple'!

(Though of course, this may sound a bit provocative... because it is also obvious that the F.B.I.'s method is much more detailed than the earlier approaches described the scientific researchers!)

Lynn, if you can not agree with this short 'summary'... will you specify where you perceive the problem?

wave

Patti Tue Apr 26, 2011 5:07 pm

A triradius is 3 separate ridges from 3 different fields that abut.
.

**Lynn** Tue Apr 26, 2011 5:17 pm

thanks for patient explanation Martijn. Where did you get all these different defnitions from?

So, while the scientific approach used to discriminate:

1 - a 'splitting', as a (asymmetric) divergence of one ridge into 2 ridges that follow same direction (no angle);
2 - a 'fork'/'Y-shape', as a (symmetric) divergence of one ridge into 2 ridges that follow the same direction (no angle);
3 - a 'bifurcation', as a divergence of one ridge 2 ridges in different directions (angle smaller than degrees);
4 - and a 'triradius' as a divergence of 2 ridge into (angle larger than 90 degrees).

**Lynn** Tue Apr 26, 2011 5:23 pm

Patti wrote:
Lynn wrote:Patti, where did you get this from?
A bifurcation's forks are usually 45 degrees or less apart.
I'm sure that's not correct.

Yesterday I quoted from FBI's info on Tented Arches.

A tented arch is different from a plain arch in that it has a triradius.

In the rules they tell us that if an upthrust from a ridge is less than 45 degrees it does not count. By deduction, that would mean the ridge rising from another ridge, is a fork or a bifurcation. Could be an appendage of some kind, dot or island? I'd see it as a short rising branch between two parallel ridges.

That ridge if at least at a 45 degree angle divides the single field into distal and proximal fields.

If that upthrust is at a 90 degree angle to the other ridges (and the other ridges are fairly horizontal in an arch) 3 fields are divided. Proximal and the ulnar and radial lateral sides.

This is how I interpreted the FBI's instructions.

hi Patti, yes I saw your post yesterday.
"an ending ridge of any length rising at a sufficient degree from the horizontal plane; i.e., 45° or more." is a definition of one type of tented arch. I don't see how you associated the 45 degree angle to bifurcations. We know for sure that bifurcations can be greater than 45 degrees.

**Martijn (admin)** Tue Apr 26, 2011 5:26 pm

Lynn wrote:thanks for patient explanation Martijn. Where did you get all these different defnitions from?

So, while the scientific approach used to discriminate:

1 - a 'splitting', as a (asymmetric) divergence of one ridge into 2 ridges that follow same direction (no angle);
2 - a 'fork'/'Y-shape', as a (symmetric) divergence of one ridge into 2 ridges that follow the same direction (no angle);
3 - a 'bifurcation', as a divergence of one ridge into 2 ridges in different directions (angle smaller than 90 degrees);
4 - and a 'triradius' as a divergence of one ridge into 2 ridges in different directions (angle larger than 90 degrees up to 120 degrees).

... that is just a specified summary of elements that are mentioned in Schaumann & Alter's book. Penrose also described the requirement for a triradius to have angles that are all larger than 90 degrees.

(Only the aspect of 'symmetry' vs. 'asymmetry' is not mentioned in Schaumann & Alter, not sure about Penrose's work - sorry, I can not specify for that aspect where I have seen it... but that detail is not important regarding the point I tried to make in that post).

Patti Tue Apr 26, 2011 5:57 pm

Lynn wrote:
hi Patti, yes I saw your post yesterday.
"an ending ridge of any length rising at a sufficient degree from the horizontal plane; i.e., 45° or more." is a definition of one type of tented arch. I don't see how you associated the 45 degree angle to bifurcations. We know for sure that bifurcations can be greater than 45 degrees.

We're told that abutting is related to two ridges meeting.
90 degrees is a right angle.
90 degrees is the minimum for the triradius angle.

We're told that there can be a nearly equal triangle formed by 3 ridges meeting at the center or 3 ridges meeting at the outer angles. The inside angle of this equilateral triangle is 60 degrees.

This 60 degrees tell us that at least 60 to 90 degrees is abutting.

I think for them to tell us that the upthrust has to be at an angle higher than 45 degrees to count indicates that at the very least, more than 45 degrees is an abutting of 2 ridges and not a splitting of ridges.

You may be correct in saying that a bifurcation can split higher than 45 degrees, but at what point/degree do you see it as no longer splitting but abutting?

Patti Tue Apr 26, 2011 6:04 pm

Martijn (admin) wrote:
Lynn wrote:thanks for patient explanation Martijn. Where did you get all these different defnitions from?

So, while the scientific approach used to discriminate:

1 - a 'splitting', as a (asymmetric) divergence of one ridge into 2 ridges that follow same direction (no angle);
2 - a 'fork'/'Y-shape', as a (symmetric) divergence of one ridge into 2 ridges that follow the same direction (no angle);
3 - a 'bifurcation', as a divergence of one ridge into 2 ridges in different directions (angle smaller than 90 degrees);
4 - and a 'triradius' as a divergence of one ridge into 2 ridges in different directions (angle larger than 90 degrees up to 120 degrees).

... that is just a specified summary of elements that are mentioned in Schaumann & Alter's book.

(Only the aspect of 'symmetry' vs. 'asymmetry' is not mentioned in Schaumann & Alter - sorry, I can not specify where I have seen it... but that detail is not important regarding the point I tried to make in that post).

Martijn, you are the only source that says a triradius is "a divergence of one ridge into 2 ridges in different directions".

**Martijn (admin)** Tue Apr 26, 2011 6:56 pm

Patti wrote:A triradius is 3 separate ridges from 3 different fields that abut.
.

Hi Patti,

My thought regarding your rather short 'definition' is that... any definition which speaks for a triradius solely about 3 'abutting ridges'... can be described as a theoretical approach only.

Because in practice one can not actually 'see' of whether in a triradius the ridges are 'abuting' ... or whether is a 'bifurcating' tradius!

And therefore, in the scientific approach we really have to consider the sizes of the angles in order to discriminate for a triradius from a 'bifurcation' or 'closure' (see Schaumann & Alter, page).

But in the F.B.I. approach these rather technical aspects (abutting ridges, bifurcating ridges, angles) become completely irrelevant.

Though only in the F.B.I. approach an 'abuting ridge' is defined very well: because the F.B.I. solely speaks about an 'abutting ridge/appendage' when two ridges make a right angle.

thinking

... I guess this is another great point in the F.B.I. they have defined all essential elements quite well, while in the earlier works many elements were mentioned... they they nearly always illustrated the tiny little elements with a visual example - but a formal definition regarding the details (angles, etc) was nearly always missing.

The F.B.I. approach is not 'misleadling', it is actually much more specified... after all in our search for elements in the words preceeding the F.B.I. book we are being confronted with certain elements of 'chaos'.

For example, both in Schaumann & Alter and in Loesch's work the word 'bifurcation' is not even mentioned in the glossary.

Fortunately, Cummins & Midlo have mentioned it - only once ( scratch

).

But now Cummins & Midlo are scoring points because they made a very interesting comment on page 31 (where the word 'bifurcation' is mentioned):

"Ocassionally a ridge may branch, forming a bifurcation or fork; if the two ridge were
considered as coursing from the opposite direction they might be described as fusing, but the common designation is based on the consideration of one ridge branching rather than of two ridges joining. Two such branches, however, may rejoin after a short course, forming an enclosure of eyelet."

( applause

)

... This explains why Cummins & Midlo do not speak about 'abutting ridges' in their definiton of a triradius, instead they only speak about 'the meeting point of three opposing ridge systems' and "three ridge systems coursing in different directions".

Later Schaumann & Alter specified that to:

"A triradius is formed by the confluence of three ridge systems... "

But it is a fundamental mistake to 'translate' their words as if Cummins & Midlo describe in their definition a situation where 3 ridges 'abutt'... my quote from page 31 in their book explains this!

wave

**Martijn (admin)** Tue Apr 26, 2011 7:06 pm

Patti wrote:
Martijn (admin) wrote:
Lynn wrote:thanks for patient explanation Martijn. Where did you get all these different defnitions from?

So, while the scientific approach used to discriminate:

1 - a 'splitting', as a (asymmetric) divergence of one ridge into 2 ridges that follow same direction (no angle);
2 - a 'fork'/'Y-shape', as a (symmetric) divergence of one ridge into 2 ridges that follow the same direction (no angle);
3 - a 'bifurcation', as a divergence of one ridge into 2 ridges in different directions (angle smaller than 90 degrees);
4 - and a 'triradius' as a divergence of one ridge into 2 ridges in different directions (angle larger than 90 degrees up to 120 degrees).

... that is just a specified summary of elements that are mentioned in Schaumann & Alter's book.

(Only the aspect of 'symmetry' vs. 'asymmetry' is not mentioned in Schaumann & Alter - sorry, I can not specify where I have seen it... but that detail is not important regarding the point I tried to make in that post).

Martijn, you are the only source that says a triradius is "a divergence of one ridge into 2 ridges in different directions".

Dear Patti... you are making the same mistake that you made yesterday: Coppock's definition clearly describes how even in the 'scientific approach' a triradius relates to a bifurcation.

Regarding Lynn's comment that Coppock only speaks about 'appear as'... well that can not be an argument to dismiss my reference to Coppock, because in analysing fingerprint be usually only can consider how the details 'appear to us'.

EDIT: And I guess that Coppock's of those words ('appear as'), is a result of the fact that he does not speak about angles in his definitions for the 'minituae'.
So, if one doesn't consider the angles... then one can only observe how the details "appear as"...!!!

(This should explain why I did not recognize Lynn's response as an 'argument'... she only described how she 'read' Coppock's words)

You should really become more aware of where exactly a triradius is described as an abuttment of three ridges... because even in the time of Cummins & Midlo that had already become a 'theory' only (see former post where I refer to page 31 in their work).

Patti Tue Apr 26, 2011 7:30 pm

I'm going to agree to disagree here.

A triradius is only 3 separate individual ridges meeting from 3 different fields. It may to some look like a bifurcation, but looking like does not make it so.

bi = 2
tri = 3

wave

**Martijn (admin)** Tue Apr 26, 2011 7:49 pm

Patti wrote:I'm going to agree to disagree here.

A triradius is only 3 separate individual ridges meeting from 3 different fields. It may to some look like a bifurcation, but looking like does not make it so.

bi = 2
tri = 3

Patti, obviously your focuss is on the 'scientific approach' only:

But even in that approach: when speaking about the 'individual ridges', you should have talked about that the 3 ridges form the borders between the three parallel ridge fields.

So, I am sorry ... but what you described literally, can not be seen in a tradius.

And I also have a problem your 'mathmatical formulas'... because both in a 'triradius' and a 'bifurcation' one can always speak about that there are 3 ridge-branches involved: see the picture below.

The TRIRADIUS in a fingerprint: how it develops, it's characteristics + a definition! - Page 5 Trirad15

The TRIRADIUS in a fingerprint: how it develops, it's characteristics + a definition! - Page 5 Trirad15

Patti Tue Apr 26, 2011 8:09 pm

The TRIRADIUS in a fingerprint: how it develops, it's characteristics + a definition! - Page 5 Drawtr10

The TRIRADIUS in a fingerprint: how it develops, it's characteristics + a definition! - Page 5 Trirad11

I prefer the scientific method, but it works perfectly with the FBI's method.

3 ridges from 3 separate ridge systems meeting at the angles or 3 ridges meeting at the corners. Abutting ridges, not bifurcating ridges. Angles not forks.

Lynn questioned the concept of 45 degrees or less being a bifurcation. But, just visualize that. We're talking the distance between two ridges. 90 degrees is straight up between the two ridges. 45 degrees would be a diagonal line between the two ridges, anything less would pretty much be parallel!

In my opinion, the angle of the widest bifurcation would be enough to allow one ridge from an opposing field to fit between its branches.

**Martijn (admin)** Tue Apr 26, 2011 8:48 pm

Patti wrote:

I prefer the scientific method, but it works perfectly with the FBI's method.

3 ridges from 3 separate ridge systems meeting at the angles or 3 ridges meeting at the corners. Abutting ridges, not bifurcating ridges. Angles not forks.

Lynn questioned the concept of 45 degrees or less being a bifurcation. But, just visualize that. We're talking the distance between two ridges. 90 degrees is straight up between the two ridges. 45 degrees would be a diagonal line between the two ridges, anything less would pretty much be parallel!

In my opinion, the angle of the widest bifurcation would be enough to allow one ridge from an opposing field to fit between its branches.

(Af first sight both the illustrations & the considerations... sound like 'theory' only; I see no practical guidelines at all)

Patti, can you find any quote in any work which explicitely describes that a 'triradius' can been seen as an abutting of three individual rigdes?

I think it will be interesting to focuss on the details of such quotes... if you can find any!?

PS. I think your comments include quite a few rethorics that I can not follow.

For example:

"3 ridges from 3 separate ridge systems meeting at the angles" ( confused

Which 'angles'?)

"or 3 ridges meeting at the corners" ( confused

Which 'corners' ?)

Again, the branches in a triradius form the BORDER between the 3 confluencing ridge systems. And only 'individual ridges' can meet and make an angle.

Plus your last sentence sounds quite unrealistic:

"In my opinion, the angle of the widest bifurcation would be enough to allow one ridgre from an opposing field to fit between its branches."

Because in the F.B.I. book we have seen so many examples of multiple 'bifurcations' where there are much more than one ridge seen in various of these 'bifurcations' - figure 26 is a clear example.

And I don't think that you are able to present any quote from any book which justifies your statement that there can manifest at most only 'one ridge' inside a bifurcation.

You also did not specify the implications of your 'theory' when there is not just one ridge... but 2 or more ridges. In the F.B.I. approach such are definitely called 'bifurcations' as well. And in the scientific approach I observed that there are only requirements regarding the angles between the ridges.... but not the number of ridges that are 'enclosed'.

So, I don't see where your 'theory' connects with what is described in the books. confused

Patti Tue Apr 26, 2011 9:33 pm

The TRIRADIUS in a fingerprint: how it develops, it's characteristics + a definition! - Page 5 Trirad12

Abutment of 3 ridge systems coursing in different directions.

**Martijn (admin)** Tue Apr 26, 2011 9:51 pm

Patti, in my request I invited you to present a quote that explicitely mentions an abutting of '3 individual ridges' I quote my request:

"Patti, can you find any quote in any work which explicitely describes that a 'triradius' can been seen as an abutting of three individual rigdes?"

I already described myself that Cummins & Midlo's quote only mentions the abutting of (1) 'ridge systems'. And how the vocabulary in Schaumann & Alter's quote more explicitely describes how these words should by 'read' - because they are speaking more detailed about a (2) 'confluence' of ridge systems; and Penrose made a likewise more detailed comment speaks about (3) 'parallel ridge fields'.

(EDIT: Can I remind you that we agreed that regarding the triradial point Cummins & Midlo's workd is slightly less detailed that the works that followed later; well their definition of a triradius is a likewise another example!)

All three statements in the books relate to the parallel ridge systems/fields that are found in INSIDE the three topographic areas. And therefore none of these statements describes an 'abutting' of three individual ridges.

Obviously, you did not 'pick-up' my earlier hint... that you would not find such a statement in the work of Cummins & Midlo (because I had already checked that work).

Sorry, your copy does not serve my request at all.
I am really looking for an explicite comment... that can in no way leave any room for 'indivual interpretation'.

Patti Tue Apr 26, 2011 10:11 pm

In earlier discussions you argued that a system or a field was related to 1 ridge and the groove on each side.

Have you now decided it is more than 1 ridge?

**Lynn** Tue Apr 26, 2011 10:30 pm

"Patti, can you find any quote in any work which explicitely describes that a 'triradius' can been seen as an abutting of three individual rigdes?"

In the text Patti posted above, it says "One of the more common types of construction....<snip> .......Three ridges radiate from a common point".

I know it says 'radiate', but it also says that all triradii are associated with abutment of 3 ridge systems coursing in different directions. So presumably these three ridges are abutting.

**Lynn** Tue Apr 26, 2011 10:46 pm

Martijn (admin) wrote:
Lynn wrote:thanks for patient explanation Martijn. Where did you get all these different defnitions from?

So, while the scientific approach used to discriminate:

1 - a 'splitting', as a (asymmetric) divergence of one ridge into 2 ridges that follow same direction (no angle);
2 - a 'fork'/'Y-shape', as a (symmetric) divergence of one ridge into 2 ridges that follow the same direction (no angle);
3 - a 'bifurcation', as a divergence of one ridge into 2 ridges in different directions (angle smaller than 90 degrees);
4 - and a 'triradius' as a divergence of one ridge into 2 ridges in different directions (angle larger than 90 degrees up to 120 degrees).

... that is just a specified summary of elements that are mentioned in Schaumann & Alter's book. Penrose also described the requirement for a triradius to have angles that are all larger than 90 degrees.

(Only the aspect of 'symmetry' vs. 'asymmetry' is not mentioned in Schaumann & Alter, not sure about Penrose's work - sorry, I can not specify for that aspect where I have seen it... but that detail is not important regarding the point I tried to make in that post).

Are these Schaumann & Alter's words, or are they your interpretation / summary Martijn? I ask because formally the words bifurcation, fork and Y shape have the same meaning in the dictionary. I have not read anywhere that a 'triradius' as a divergence of one ridge into 2 ridges in different directions

Sponsored content