1. DISCLAIMER.

2. ABSTRACT, APIII, #376.

 Triple-Spiked Zones in Cell Surface Tessellations:
 Model for Malignant Growth.
 G. William Moore, MD, PhD (George.Moore4@va.gov) [1,2,3];
 Raimond A. Struble, PhD [4]; Lawrence A. Brown, MD [1,2];
 Grace F. Kao, MD [1,5]; Grover M. Hutchins, MD [3].
 Pathology and Laboratory Medicine Service, Veterans Affairs
 Maryland Health Care System, Baltimore, MD [1];
 Department of Pathology, University of Maryland Medical System,
 Baltimore, MD [2]; Department of Pathology, The Johns Hopkins
 Medical Institutions, Baltimore, MD [3]; Department of Mathematics,
 North Carolina State University, Raleigh, NC [4]; and Department
 of Dermatology, George Washington University School of Medicine,
 Washington, DC [5].
                   http://www.medparse.com/triplspk.htm 
                                 
 Content: Tumors of surface epithelium are among the most common human
 malignancies.  In benign surface epithelium, the cell surface exhibits
 a regular, repeated packing of cells, or tessellation, resembling
 a collection of equal cylinders resting side-by-side. Malignant
 transformation involves variably-sized cells, a disorganized surface,
 and the tendency to invade surrounding tissues.
                                     
 Technology: Ordinary and synthetic geometry.
                   
 Design: Mathematically, a tessellation is a periodic tiling of the plane
 by polygons, or space by polyhedra.  A cell tessellation is a surface
 of nearly-circular cell-apices.  The triple-spiked zone is the region
 inside each triple of tangent circles. Each tangent cell-triple has
 an inscribed circle, tangent to the triple-spiked zone, that forms
 a circular communication zone among the original tangent circles.
                                    
 Results: It is demonstrated that a cell-triple communicates maximally
 if and only if all three cell-radii are equal.
                                               
 Conclusion: Malignant surface cells are characterized by more size variation
 and less balanced packing. In this model, unequal cell size and cell
 miscommunication are geometric features of the same underlying process.
 Therapy for the communication process might possibly control the cell-size
 process. Mathematical models can be used to explore alternatives to classical
 hypotheses in pathology, and explore general paradigms.

3. OUTLINE OF EPOSTER.

2282.     9930.

2280.     9931.

2322.

2522.     2384.

2321.

2251.     2383.

2383.

2384.

2522.     2384.

3. Re-proved by Giovanni Ceva in De lineis rectis (1678).
2761.

4. The intersection point in the triangle is called the Gergonne point.

---

18. Proof of Ceva's Theorem. Part A.
2762.

Therefore, ace=bdf, Q.E.D.

---

18.1. Proof of Ceva's Theorem. Part B.
2763.

Since (a'+b') = (a+b) = YZ, it follows that a'=a and b'=b. QED.

---

19. Centers-lines.
2385.

Satisfy Ceva's Theorem.

---

20. Tangent-lines.
2387.

---

21. Inscribed radius.
2389.

---

22. Inscribed radius in triangle.
2392.

---

23. Perpendiculars in triangle.
2393.

---

24. Perpendiculars in triangle.
2395.

---

25. Alignment Interpretation.

1. Malignant/dysplastic cell surfaces are MISALIGNED:

2384.     2383.

---

26. Alignment Interpretation: Critique.

1. For very different circle-sizes, distance between the center-tangent and tangent-line is small.
2. Obtuse triangle case for large distance between intersections.
3. Amount of disorganization evoked by malignant cells is not great.
4. Small disorganization apparently sufficient for tissue-invasion.

---

27. Discussion.

1. Alignment interpretation is problematic.
2. Large cell size differences must account for very small misalignments.
3. New interpretation: inscribed circle within the triple-spiked zone.
4. Inscribed-circle-to-triple-spike-zone maximized for equal-radius circles.
5. Communication zone among tangent cells, maximized for normal cells.
6. Communication zone becomes with dysplastic, malignant degeneration.

---

28. Conclusion.

1. Malignant surface cells have more size variation, less balanced packing.
2. Unequal cell size and cell miscommunication:
3. Geometric features of same process.
4. Therapy for the communication process
5. Might control the cell-size process.

---

29. Why Tissue Geometry?

3. Open area of geometric research in mathematics.

---

30. Mathematical models.

1. Explore alternatives to classical hypotheses.
2. Explore general paradigms.

---

INTRODUCTION AND NOTES.

The great majority of human malignancies involves uncontrolled growth over cell surfaces: epidermal, mucosal, and serosal. A cell surface, viewed en face, resembles a regular tiling on the plane, or tessellation (Latin: tessella = tile), of tangential, nearly-circular cell apices:

2282.     9930.

As these cells undergo dysplasia and eventually malignant transformation, the individual cells show greater size variation, and the cellular distribution over the tessellation becomes more haphazard. Traditionally, it has been assumed that cell growth and variation is the primary biological process, and surface irregularity is secondary. Our laboratory proposed an alternative hypothesis, namely, that tangent-cell interactions are the primary process, using a mathematical theorem that tangent circles have equal radii if and only if their center-center and center-tangent lines are exactly aligned, i.e., share a common intersection point. In this report, we propose a third hypothesis: that tangent-cells communicate with one another through triple-spiked zones of close contact.

Mathematically, three tangent circles on a plane form a triple-spike zone, or region inside the three tangent points:

2251.     2580.

The ratio of the inscribed circle area, to that of the entire triple-spike zone is maximized for equal-radius circles, namely, the prototype for normal cell growth.

2522.     2581.

Thus the region for cell-cell potential contact and communication is maximized for normal cell growth.

The mathematical proof involves a triangle theorem, in which the vertices of any triangle coincide with the centers of three tangent circles, and the vertices of an equilateral triangle coincide with the centers of three equal-radius tangent circles.

Discussion: In our previous interpretation of equal-radius circles, we suggested that cell-cell control might involve an optimal alignment of tangential normal cells, which corresponds to equal-radius circles. A possible flaw in this interpretation is that slight variations in the center-center and center-tangent lines corresponds to huge variations in cell radii. Our current interpretation is that cell-cell communication occurs within the triple-spike zone. As the volume for this communication diminishes, the cell sizes become more haphazard, and malignant degeneration follows.

In addition to the mathematical proof for the triangle theorem, and its corollaries for tangent circles, it is instructive to vary the parameters for the three circles, and observe the consequences. A public-domain computer script previously published by our group has been modified for this purpose.

Proof of Ceva's Theorem.......

Another interesting features of the triangle theorem is that one may construct cascades of triangles within triangles, and therefore, cascades of circles within circles. As with fractals applied to biological processes, this is a rather abstract operation, since cells do not become either arbitrarily small or arbitrarily large. Still, it supports the idea that the mathematical model is robust over a range of cell sizes and configurations.......

We have modified a previously-published, public domain computer script for drawing computer blank-white graphics for synthetic geometry simulations, involving triangles and circles. The program is written in Perl (eventually, Ruby), and uses widely available tools, such as Microsoft® Paint® and Microsoft® PictureManager® and eventually, Microsoft® Excel®. Foreseeably, one could formulate an animation program to produce cinematic simulations.

For example, here is a collection of circles, all with center (256,192), and radii 90, 80, 70, 60:

0090.   0080.   0070.   0060.

We begin with the default, blank-white screen displayed by Microsoft® Paint®, which is 512 pixels (dots) wide and 384 pixels tall, a total of 589,824 pixels. This is approximately the same resolution as an ordinary television screen.

To keep the program simple, we shall exclusively use this file size and format. The actual file size is 589,824+54=... bytes. The first 54 bytes contain formatting information required by the .bmp file format. In particular, bytes 19, 20, 23, 24.....

The Perl script works by reading in a blank screen, reading out the first 54 bytes, and closing the blank screen. The script checks to verify that bytes 19, 20, 23, 24, are correct. The remaining pixels are 3 bytes apiece, one for each of red-green-blue. Maximum bright: 255. Maximum black: 0. The Perl script sets the array $pxl[512][384] to =1 for black, =0 for white.....

Last year's presentation began with the observation that an en face view of a normal cell surface may be modeled as a collection of tangent cylinders with equal radii. Dysplastic and malignant cell surfaces show unequal radii, and disorganization of the cell tiling, or tessellation. Lines drawn between neighbor-cell-centers and cell-center-to-tangent-points have a common intersection point (i.e., are in exact alignment) if and only if the cells have equal radii. We concluded that misalignment of surface cells might be associated with dysplastic and malignant degeneration.

This interpretation is problematical, because large differences in cell size are necessary to account for only small misalignments. It is hard for me to believe that these small misalignments are the driving force behind large cell size differences. We stated this problems parenthetically in the original presentation last year. This year, we suggest a new interpretation. Instead of center-center and center-tangent alignments, we focus upon the inscribed circle within the triple-spiked zone, i.e., the region inside each triple of tangent circles. The inscribed-circle-to-triple-spike-zone relationship is MAXIMIZED for equal-radius circles:

2251.    2580.

2522.    2581.

We suggest that the area ratio of the two regions forms a COMMUNICATION ZONE INDICATOR among tangent cells, that is maximized for normal cells, and becomes smaller in conjunction with dysplastic and malignant degeneration.

Proposition 0. There exist center-lines.

Proposition 1. There exist center-to-tangent-lines.

Proposition 2. There exist tangency-lines.

Proposition 3. Center-to-tangent-line intersection point and tangency-lines intersection point are concurrent if and only if triangles are equilateral.

Proposition 4. Edges of any triangle contain tangency-points of three externally tangent circles.

Proposition 5. .........

2274.
When the radius of this first circle is appropriate, you can draw circles with a compass, whose centers are B and C, and whose circumferences intersect the triangle at points b and c.

INTRODUCTION. Tumors of cuboidal or columnar epithelium are among the most common human malignancies, including the most common types of cancers of the skin, lung, colon, breast, prostate, uterus, kidney, and urinary bladder. Benign cuboidal or columnar epithelium is an orderly arrangement of cells overlying a flat or curved surface of supporting tissues. Progression to dysplasia and carcinoma involves the apparently independent processes of increased variation in cell size, variation in nuclear ploidy, and disordered arrangement of cells, leading to invasion of underlying tissues. Invasion of underlying tissues is regarded as the defining feature of malignancy.

We propose a mathematical model in which cell size variation and disorder are equivalent geometric properties. Therefore, diagnosis and prognosis of one feature mirrors diagnosis and prognosis of the other feature. Cell size may be easier to measure and control; whereas disorder and invasion are the eventual/ultimately destructive features of cancer.

The nubbin of this manuscript is that, on a sheet of cells laid out contiguously, every cell-triple (except at the boundary of the sheet), has six lines that may be drawn through it. They are: a line from the center of each circle to the tangent-point for the other two circles (center-tangent-intersection); and a tangent-line for each pair of circles (tangent-line-intersection).

In the branch of mathematics known as topology, any set of planar objects that covers the plane is called a tiling. Specifically, a set of planar objects is a tiling if and only if the closure of its union equals the plane. If the set is repetitive, then it is called a tessellation (latin: tessella=tile). a surface of cuboidal or columnar epithelium may be likened to a collection of tightly-packed cylinders, i.e., cylinders that touch (are tangent to) one another. If the cylinders are all equal, then only (1 - π/4) ~ 0.2146 of the surface remains to be filled, in order to complete the tessellation. Cells may signal one another by their relative positions with respect to their neighbors. In benign cuboidal or columnar epithelium, except at the edge, all cells touch one another exactly as they are touched, and thus receive the same signals as those that they send. Cancer may be defined as unbounded cell growth. benign cells are signalled to stop proliferating when they reach a certain limit, such as wound-healing, scar formation, ulcer re-epithelialization , ......

A sheet of cuboidal or columnar epithelium resembles a collection of closely packed cylinders. The cell-surface is the apex; and the cell-attachment to underlying tissues forms the base. Typically, the cell-nucleus is located at the base of the cell, whereas a majority of cell-cytoplasm is located at the apex of the cell.

Viewed en face, the cell-surface appears as a honeycomb; viewed in cross-section, the cell-surface appears as a picket fence.

It has long been recognized from tissue culture studies that malignant cells are less cohesive to one another than normal cells, and form a less organized surface. Variable nuclear ploidy is also a widely-observed property of malignant cells.

The lines intersect at a common point, i.e., the cell-triple is balanced, if and only if all three cell-radii are equal. Since benign cells have essentially equal radii, but premalignant (i.e., dysplastic) and malignant cells do not, perhaps this line-intersection property is an interesting property that disappears in malignant degeneration. The mathematical model simply asserts that the common-intersection and the equal-radii properties are mathematically equivalent. If you have one property, then you have the other; and vice versa.

The implications for therapy are interesting: If you could control one process, then you have controlled the other, since the processes are mathematically equivalent.

When you draw a few sample pictures (see below), the theorem doesn't seem very profound. Any bright high-school plane geometry student, and probably Euclid himself, could demonstrate that the if part of the theorem is true. This simplicity might lead the unwary to suspect that the entire theorem is nothing more than a high-school extra-credit problem.

However, proof of the only if part of the theorem is quite involved and tricky, as we shall see. At the very least, a knowledge with infinite series and limits is required, which was way beyond Euclid. .............

Struble's manuscript begins with the following three assertions. For any three non-intersecting circles, tangent to one another:

(i) the three line-segments drawn from the centers to the opposing tangency-points (i.e., center-tangent-intersection) meet at a common point,

(ii) the three tangency-lines (i.e., tangent-line-intersection meet at a common point,

and (iii) these two meeting-points are coincident only if the three circles have equal radii. (Only then are they coincident with the center of the largest circle that can be embedded within the triple-spiked region which they form, as in Figure 2.)

That is, Struble proves that, for any tangent-circle-triple, (i) the center-tangent-intersection exists; (ii) the tangent-line-intersection exists; (iii) all six lines intersect at a common point only if all three circle radii are equal.


2251.


The IF part of the proof is an exercise in high-school geometry:


2212.
Without loss of generality, construct three tangent circles of radius 1, where centers of the two lower circles, left and right, rest on the abscissa (x-axis). Their tangent-point is the origin, (0,0), and their tangent-line is the ordinate (y-axis):


2213.
We then draw the equilateral triangle, with base-segment along the centers of the two lower circles, apex at the center of the upper circle, and each side equal to 2:


2214.
The tangent-line, TL, for the lower two circles is the ordinate, which coincides with the line-segment that joins the upper-center to the lower tangent-point, TP.

A similar construction, using the other two sides and the Pythagorean theorem, determines that the intersection of all six points lies at coordinates: (0, 1/√3).

The ONLY IF part of the proof is an advanced problem. For his proof, Struble reframes the problem as a problem with triangles. The first part of Struble's proof is based upon the following figure where circles possess very different radii, approximately 8:1 to 3:1):


2259.
That is, you can draw a circle with a compass, whose vertex, A, has two points along its adjoining sides, i.e., points b and c, such that Ab equals Ac. You can then draw a circle with a compass, whose center is vertex A, and whose circumference intersects the triangle at the points b and c. A perpendicular drawn with respect to the triangle, drawn at those intersection-points b and c, will be tangent-lines to the circle, and satisfy the earlier definition of tangent-line-intersection. Struble's proof is an application of Ceva's Theorem (1678).

---

Cuboidal or Columnar Epithelial Tumors.

4. Seen in many drawings by M. C. Escher.

---

Cell Surface Tessellation.

9264.

---

Cross-section: Picket Fence.

2286.


9929.

---

En-face: Honeycomb.

2282.


9930.

---

En-face: Malignancy.

2280.


9931.

---

Nearly-Circular Cell Apices.

---

Cell Surface Tessellation.

Cell-triple is balanced if and only if these six lines meet at a single intersection point.

---

Tangent-line. Center-opposite-line.

2521.

---

Balanced/Unbalanced Cell Triples.

2522.


2521.

---

Mutually Tangent Circle Theorem.

2214.

Proof of Only-If: Advanced problem.

---

Proof: If.

2214.

By Pythagorean Theorem.

---

Proof: Only If. Construct points D, E, F.

2528.

---

Proof: Only if. Point D.

2525.

---

Only If, Part (i). Point D.

2525.

There exists a unique point D at the intersection of center-opposite-tangent lines.

---

Proof: Part (i). Point D.

2540.

Thus Ca=cA and d=a.

---

Proof: Only if. Point E.

2526.

---

Only if, Part (ii). Point E.

2526.

There exists a unique E at the intersection of tangent-lines

---

Proof: Part (ii). Point E.

2530.

Paired sets of congruent triangles, i.e., CaE = CbE, AcE = AbE, BaE = BcE.

---

Proof: Only if. Point F.

2527.

---

Only if, Part (iii). Point F.

2527.

There exists a unique point F and internal circle radius r such that center-to-F minus r for an external circle equals the radius of the external circle.

---

Proof: Part (iii). Point F.

2531.

Points A, B, and C pass through the center of the internal circle, F.

---

Proof: Only If. Points D, E, F.

2538.

---

Proof: Part (iv). Points D, E, F.

2538.

Points D, E, F are coincident only for equilateral triangles.

---

Points D, E, F are collinear.

2528.

---

Summary: Mutually Tangent Circle Theorem.

2521.


Tangent-lines and center-opposite-lines intersect at a common point if and only if all three cell-radii are equal.

---

Struble Triangle Theorem.

2531.

(ii). There exists a unique interior point E, for which three line segments emanating from E to the points a, b and c are perpendicular to the edges of the triangle.

---

Struble Triangle Theorem.

2531.

8. Mathematical models can be used to propose alternatives to classical hypotheses in pathology.

---

Larry Brown has briefly vetted this idea, and he points out that actual cells on a surface are not quite circles, but rather, more-or-less-hexagons. If the Struble tangent-circle theorem only applies to circles, then it is, strictly speaking, irrelevant to cellular growth. We don't regard this as a big issue. It seems to me that you could find the area and center-of-gravity of each nearly-circular cell, and liken this to the circle of the same area and center-of-gravity. You might have to do a little more jostling to make everything fit together on a tiling or tessellation.

One of the amazing things about Struble's tangent-circle construction is that, even for wildly differing circle-sizes, the distance between the center-tangent-intersection and the tangent-line-intersection isn't very great. In fact, one needs to view the obtuse triangle case, to see any appreciable distance between the intersections. See:


2251.
This means that the amount of disorganization/imbalance evoked by malignant cells (if, indeed, this mechanism is appropriate) is not great, but is apparently sufficient for the next (and defining) step of malignancy, namely, tissue-invasion.

Furthermore, essentially all cells, even malignant cells, have a narrow size limit, from perhaps 9 nm minimum (small lymphocytes) to 200 nm maximum (malignant syncytiotrophoblasts, which might not even legitimately qualify as single cells). (Erythrocytes (7 nm) and platelets (3 nm) don't count as full-fledged cells, since they have no nuclei, and cannot reproduce.) Struble's manuscript shows that, even for a 20-fold difference of radii in a cell-triple, the center-tangent-intersection-point and the tangent-line-intersection-point are quite close.

We have used the terms tiling and tessellation somewhat sloppily. According to Mathworld on the internet, a TILING is a plane-filling arrangement of plane figures, or its generalization to higher dimensions. Topologically, a tiling is a collection of disjoint open sets, whose closure covers the plane. Wang's conjecture (1961) states that if a set of tiles tiles the plane, then they could always be arranged to do so periodically. A periodic tiling of the plane by polygons or space by polyhedra is called a TESSELLATION (Latin: tessella=tile).

1. Malignant tumors of surface cells, including tumors of ectodermal and endodermal origin, are the most common human malignancies, accounting for over twenty million new cases of malignant tumor annually worldwide.

2. A normal cell surface may be regarded as a collection of cylinders, packed side-by-side, where each cylinder is about 10µ in diameter and 20µ in height. Typically, the cells may be viewed over the surface (en face) or in cross-section ("picket fence").

Normal surface-cells, cross-section:
9929.


Normal endocervical cells, cross-section:
2286.


3. On a normal cell surface, viewed en face, the cells appear as approximately equi-radius circles, densely packed together in a honeycomb arrangement, over a nearly planar surface.

Normal surface-cells, en-face:
9930.


Normal endocervical surface-cells, en-face:
2282.


4. In malignant surface tumors, cells have a broader distribution of radii, less regular packing, and protrusions from the surface, either papillomatous (above-surface) or acanthomatous (below-surface).

Malignant surface cells, en-face, varying sizes:
9931.


Malignant surface cells, en-face, varying sizes:
2280.


5. We propose a mathematical model for a normal cell-surface tiling, or tessellation, in which contiguous cells sense their relationships to one another, and stabilize/balance their cell-radii, based upon intercellular relative positions of cells.

6. We propose that each cell LOOKS AT its immediate two, neighboring cells, to form a CELL TRIPLE, in a topological relationship. Each cell has a CENTER and a TANGENCY LOCUS with its two, neighboring cells:

2201.


In this illustration, there is a large cell (A, top), a medium cell (B, lower left), and a small cell (C, lower right), all viewed en face.

2202.


The centers of these cells are a (large), b (medium), and c (small), respectively:

2203.


Tangency-points (◅) are labeled TPab between the large and medium circle.......

2206.


Tangency-lines (——) are labeled TLab between the large and medium circle.......

2209.


7. The CELL NUCLEUS is located at the center of each cell, and the nucleus SENSES the TANGENCY LOCUS at its two neighboring cells.

8. Each TANGENCY-LOCUS is, itself, a TANGENCY-POINT; and projects out as a TANGENCY-PROJECTION-LINE. For each cell facing two other cells, there is a center-to-tangency-point line (CENTER-TANGENT-LINE) and a TANGENCY-PROJECTION-LINE.

9. Cells are IN BALANCE when all six lines (i.e., three center-tangent-lines and three tangency-projection-lines) intersect as a common point.

10. This formulation is analogous to the the so-called FRENCH FLAG PROBLEM in cellular biology. That is: if you begin with a single, vertical line of cells (i.e., the flagpole), and the cells divide and proliferate rightward, what chemical signal is sent between the cells that tell a line of cells to be BLUE, WHITE, or RED (in order from left to right, the colors of the French Flag, or tricoleur). Each cell can only send signals to its immediate neighbors, a sort of Markov/topological constraint.

11. Normal cells are in optimal balance, if and only if tangent-line-intersection equals center-tangent-intersection. Then their cell-radii are equal.

12. Struble Tangent Circle Theorem states that tangent-line-intersection equals center-tangent-intersection if and only if the cell-radii are equal.

13. Malignant surface cells are characterized by more size variation and less balanced packing. We suggest that these processes correspond to a loss of look-across-line and tangency-line coordination.

The U. S. National Cancer Institute (USNCI) of the U. S. National Institutes of Health (USNIH) estimates that there are one million new cases of skin cancer, the most common malignancy, annually in the USA:

http://www.cancer.gov/cancertopics/wyntk/skin

The next leading malignancies: prostate, breast, lung, colon, have on the order of 10⁵ new cases annually in the USA. The most common malignancies from these organs far and away/predominantly involve the ectodermal/endodermal embryonic cells. The USA has one-twentieth of the world's population, so, conservatively estimated, there are twenty million new malignancies annually worldwide. [Cancer is predominantly a disease of older persons, so that in areas of the world with short life-expectancies, the incidence of cancer may be less.]

A plane-filling arrangement of plane figures or its generalization to higher dimensions is called a TILING. Topologically, a tiling is a collection of disjoint open sets, the closures of which cover the plane.

Wang's conjecture (1961) states that if a set of tiles tiles the plane, then they could always be arranged to do so periodically. A periodic tiling of the plane by polygons or space by polyhedra is called a TESSELLATION.

See: http://mathworld.wolfram.com/Tessellation.html

French Flag Problem. "The pièce de résistance of the range, however, is a little genetic joke. Embryologists in the past spent a lot of time worrying about something they called the French flag problem. This was how simple chemical gradients in an embryo could result in complex body patterns. Their idealised version of this question was to ask how two signal chemicals, starting from opposite ends of an embryo, could result in a pattern that looked like the French flag (ie, three different coloured vertical stripes). The French flag problem was solved by the discovery of so-called "hox" genes that control embryonic development, and much of the early hox work was done in zebra fish. Dr Fril has used this knowledge to create a FrenchFlag fish, with a blue head, a white body and a red tail, and hopes to follow it soon with one depicting the stars and stripes. As a wag once put it "one man's fish is another man's poisson." See:
http://www.economist.com/diversions/displaystory.cfm?story_id=348179

Prove that if three cevians of a triangle are concurrent, then extensions of the three lines passing through their feet intersect extensions of the edges of the triangle along an external line. Conversely, if extensions of the edges of a triangle intersect at three points of an external line, then there exist concurrent cevians possessing feet which are collinear with those for intersections. (Appropriate versions hold interesections at infinity.) Upon defining a suitable topology for the space of external lines, show that this one-to-one association becomes topological. Show that whenever an interior (concurrent) point tends to a (non-vertex) edge-point, the corresponding exterior line tends to that edge-line. Show that any continuous internal curve leads to an external curve defined by a continuously-turning tangent. Indeed, all the geometry of interior points in a triangle is reflected by the geometry of exterior lines.

This problem is a bit of a project in synthetic geometry, with the show part involving angles and deviations from a fixed interior point (say, centroid), for example.


2549.


The one-to-one correspondence that you described to me last night, between points and topological neighborhoods within a triangle and lines and topological neighborhoods outside the triangle, if provable, would be quite a stunning result. It reminds me of Descartes's one-to-one correspondence between the points of Euclidean geometry and analytical geometry, which nearly doubled the number of theorems in 17th century mathematics, by making every theorem about Euclidean geometry a theorem about analytical geometry, and vice versa. (We say NEARLY doubled, because there must have been at least some common theorems of geometry and algebra known before Descartes.)

Dr. Larry Brown pointed out an associated passage in the New Testament, in which OUR LORD bestows the Keys to the Kingdom upon Saint Peter:

Matthew 16:18-19. And I say also unto thee, That thou art Peter, and upon this rock I will build my church; and the gates of hell shall not prevail against it. And I will give unto thee the keys of the kingdom of heaven: and whatsoever thou shalt bind on earth shall be bound in heaven: and whatsoever thou shalt loose on earth shall be loosed in heaven.

Let Euclidean geometry be GOD and analytical geometry be Saint Peter, and bind/loose properties correspond to theorems, and there you have it, the Biblical prophecy of analytical geometry, spoken by Jesus Himself. (Euclidean geometry predates Jesus.) This passage is considered very significant to Roman Catholics. It is engraved, in two-meter-high letters in Latin and Greek, on the western ceiling of the Sistine Chapel in the Vatican, and is believed by Roman Catholics to represent OUR LORD's commission to Saint Peter for the special status of the Pope as the Vicar of Christ.

Dr. Larry Brown does not claim to be the first person to point out this association between Descartes and Mt 16:18-19, but he knows of no prior source. Do you?

Dr. Jules Berman pointed out to me that the world's largest cell is an unfertilized ostrich egg, circa 10 cm. Compare this to bone marrow reticulocytes, circa 10 µ, giving a ratio of 100,000:1 for a pair of tangent cell-circles. The designation of an ostrich-egg as a single cell (just as the demotion of Pluto from planetary status) is scientifically somewhat arbitrary. The ostrich egg consists of a single cell nucleus, about 8 µ, at the periphery of a gigantic vacuole, consisting of egg-yolk and egg-white, or albumen. A vacuole is a bubble of non-living chemical within a cell, such as a fat-vacuole inside a fat-cell, or adipocyte. Furthermore, since an ostrich egg lives in isolation, with no surrounding cells, strictly speaking, it is not covered under our definition of cell tessellation, which requires a minimum of at least three tangent circles. Likewise, blood cells, which live as isolated boats in the fluid blood, are not strictly included under our definition of cell tessellation. Thus our theory makes no claims about hematogenous malignancies, such as leukemia.

http://www.newton.dep.anl.gov/askasci/mole00/mole00128.htm

The cell tessellation paper generated enough interest in Vancouver that I hope we can keep the idea alive for next year's meeting in Pittsburgh. Perhaps these considerations about tangent circles, triangles, and topological mappings, could evolve into further properties about surface cancer-cell relationships. An idea something like: the behavior of certain points in the triple-spike region (or equivalently, the corresponding triangle) influences the growth/cancer properties of the tessellation as-a-whole.

There is a small, obscure literature in this field, dating from the 1960s, which some National Cancer Institute researchers (especiallly Dr. Judith Prewitt) and even Dr. Mary B. Williams (when she was a graduate student at the University of London, in collaboration with the prominent tissue-culture researcher, Prof. Louis Wolpert) dabbled in, which I need to look into further.

Finally, it is interesting to note that the largest human cells include the human ovum and the syncytiotrophoblast, a cell formed by the embryo in a pregnant woman, whose job it is to dig into the lining of the uterus and seize a blood supply from the mother. More than one researcher has noted the similarity between pregnancy and invasive malignancy. In pregnancy, the embryo increases maternal heart output by 40%, and in the olden days of rheumatic heart disease before penicillin, would cause heart failure and sometimes death of the mother. Tumors may do the same sort of thing. The difference is: after nine months, the pregnancy is over.

---

REFERENCES.

1. Wolpert L.
Role of diffusible gradients in regeneration.
Dev Biol. 1973 Feb;30(2):concl4-5.
PMID: 4703683.
PubMed Entry

2. Wolpert L.
Positional information and pattern formation.
Curr Top Dev Biol. 1971;6(6):183-224. Review.
PMID: 4950136.
PubMed Entry

3. Wolpert L.
From engineering to positional information to public understanding. An interview with Lewis Wolpert. Interview by James C Smith.
Int J Dev Biol. 2000;44(1):85-91.
PMID: 10761852.
PubMed Entry
"Lewis Wolpert is one of the most influential developmental biologists in Britain and the world. His concept of positional information, developed 30 years ago, changed the way we thinkabout pattern formation in the embryo and allowed new generations of molecular developmental biologists to frame their questions in a way that would give sensible answers...."
Wolpert's work involved an informal collaboration with Dr. Mary B. Williams, who was also the advisor for Dr. Moore's PhD Thesis.

4. Chandler RE, Meyer CE, Rose NJ.
Eudoxus meets Cayley.
Amer Math Monthly. 2003;110:912-927.

5. Coxeter HSM, Greitzer SL.
Ceva's Theorem.
§1.2 in: Geometry Revisited. New Mathematical Library.
Washington, DC: Math Assoc Amer. 1967;:4-5.
ISBN: 0883856190, 207 pages.

6. Durell CV.
Modern Geometry: The Straight Line and Circle.
London: Macmillan. 1961::40-41.
ASIN: B0007JDUWE, 145 pages.

7. Grünbaum B, Shepard GC.
Ceva, Menelaus, and the Area Principle.
Math Mag 1995;68:254-268.

8. Honsberger R.
Ceva's Theorem.
§ 1.2 in: Episodes in Nineteenth and Twentieth Century Euclidean Geometry. New Mathematical Library. Honsberger R, McAsey MJ, Benjamin A, Lange LH, Berzsenyi G, Saul ME, Guy RK, Straffin PD, Honsberger RA, Weiss JW, eds.
Washington DC: Math Assn America. 1996;:136-138.
ISBN: 0883856395, 188 pages.

9. Pedoe D, Watkins W, Alexanderson GL, Choudhury D, Firey WJ, Kalman D, Lang E, Merris RL, Nunemacher JL, Parker EM, Pedersen JJ.
Circles: A Mathematical View. Second Edition.
Washington DC: Math Assn America. 1997;:.
ISBN: 0883855186, 137 pages.

10. Wells D, Sharp J (illustrator).
The Penguin Dictionary of Curious and Interesting Geometry. Penguin Science.
London: Penguin Books. 1992;:28-29.
ISBN: 0140118136, 285 pages.

11. Weisstein EW.
Tessellation.
MathWorld: A Wolfram Web Resource.
http://mathworld.wolfram.com/Tessellation.html
Site last tested: 5/24/2006.

12. Weisstein EW.
Topology.
MathWorld: A Wolfram Web Resource.
http://mathworld.wolfram.com/Topology.html
Site last tested: 5/24/2006.

13. Weisstein EW.
Set.
MathWorld: A Wolfram Web Resource.
http://mathworld.wolfram.com/Set.html
Site last tested: 5/24/2006.

14. Weisstein EW.
Set Theory.
MathWorld: A Wolfram Web Resource.
http://mathworld.wolfram.com/SetTheory.html
Site last tested: 5/24/2006.

15. Weisstein EW.
Set Union.
MathWorld: A Wolfram Web Resource.
http://mathworld.wolfram.com/Union.html
Site last tested: 5/24/2006.

Wolpert L.
The French flag problem: A contribution to the discussion on pattern formation and regulation.
In: Waddington CH, ed. Towards a Theoretical Biology. Edinburgh University Press, Edinburgh. 1968;:125-133.

Wolpert L.
The French flag problem........ See:
http://www.economist.com/diversions/displaystory.cfm?story_id=348179

Ostrich egg has one cell:
http://www.newton.dep.anl.gov/askasci/mole00/mole00128.htm

---

Last updated: 8/27/2007, by G. William Moore, MD, PhD.