Four Pi Appears Only When the Source Is a Point ── Raise the Dimension of the Source by One and Four Pi Becomes Two Pi While One Over r Becomes a Logarithm: Sources in Living Tissue Are Often Not Points ── [Paper 258]

Abstract

Paper 02 showed that a pure solid angle of four pi appears in an inverse-square field. This paper asks when that four pi does not appear; the answer is the moment the source stops being a point. Raising the dimension of the source by one alone turns four pi into two pi and turns the one-over-r fall into a logarithm. No new mathematical theorem and no new law is claimed. Scope of this paper (scope note): no new mathematical theorem and no new law is claimed. The potentials of point, line and plane sources, the volume-conductor approximation, and Krogh's cylinder model are all standard. No electrophysiology is built ── what is used is three solutions, one square root, and a conversion of units. The origin of the four pi is not discussed ── Paper 02 treats that, and this paper treats only the converse, the condition under which that four pi disappears. Inverse problems are not entered ── Paper 99 asks whether the interior conductivity can be recovered from boundary data and Paper 159 separated the senses of ill-posedness, while this paper looks in the forward direction alone, asking which solid angle appears once the shape of the source is given. Capture rates are not treated ── Paper 249 treated the Smoluchowski capture rate for a sphere, in steady state, with an absorbing boundary, whereas this paper takes a cylinder, includes a consumption term, and asks not for a capture rate but for a reach. No physiological claim is made ── no account is offered of why capillaries are arranged as they are; an agreement is reported and no cause is asserted. Tissue is treated as isotropic and homogeneous ── real tissue is neither, and the numbers are values for seeing orders of magnitude rather than measurements on a particular tissue. The source is treated as an infinitely long cylinder ── at finite length the ends return towards four pi, and what is claimed is only that in the infinitely long limit the four pi does not appear. The relation to earlier papers. Paper 02 showed that a pure solid angle of four pi appears in an inverse-square field; this paper writes the domain of that four pi, namely that it appears only when point source, three dimensions and isotropy hold together. Paper 249 showed that the four pi that catches molecules and the four pi that spreads a field share a root; this paper sets out the cases where the same four pi does not appear, since confirming a shared root and confirming a domain are two halves of one thing. Paper 01 showed from four directions that n equals two is unique; what this paper moves is the dimension of the source rather than the exponent, a different axis. Paper 199 wrote that what decides whether you come home is the exponent in the denominator; the fourth section here likewise sets out the form of the denominator, but asks about the fall rather than about recurrence. Paper 112 counted resolution as one of six distinct roots; measurement resolution is not treated here. First, for a point source the four pi stands in the denominator. With a current of one microampere in a medium of conductivity 0.33 siemens per metre, the potential is 48.2288 microvolts at five millimetres, 24.1144 at ten, 12.0572 at twenty and 4.8229 at fifty. Doubling the distance halves the potential exactly, and the four pi in the denominator, 12.566371, is the whole solid angle of the sphere itself. Second, this is the core. Make the source an infinitely long line and the four pi disappears, leaving two pi and a logarithm. With a current of ten to the minus four amperes per metre the prefactor is 4.822877e-5 volts, and the potential difference is that prefactor times the logarithm of the ratio of distances: 33.4296 microvolts for a ratio of two, 111.0508 for ten, and 222.1017 for a hundred. Raising the dimension of the source from zero to one alone halved the solid angle in the denominator, from 12.566371 to 6.283185. Third, the fall itself reverses direction. Taking one millimetre as the reference for the line source and comparing over the same distances, the point source gives 120.5719 microvolts at two millimetres falling to 4.8229 at fifty, a factor of 25.0, while the line source gives 33.4296 rising to 188.6721, a factor of 5.64 in the other direction. The same distance is being travelled and the directions are opposite. One cannot say that a field weakens with distance without first writing down the shape of the source. The separator is raising the dimension of the source by one, and nothing else: neither the medium, nor the current, nor the number of dimensions of the space has been moved. Fourth, three sources are called over. A point, of dimension zero, gives four pi equal to 12.566371 and falls as one over r. A line or cylinder, of dimension one, gives two pi equal to 6.283185 and falls as a logarithm. A plane or layer, of dimension two, gives two pi and does not fall at all. Each rise in the dimension of the source flattens the fall by one step. The four pi appears in the first row alone, and written out the condition has three items, point source, three dimensions and isotropy, any one of which suffices to remove it when lost. Fifth, sources in living tissue are often cylinders. Oxygen diffuses out of a capillary and is consumed on the way, with a reach equal to the square root of twice the diffusion coefficient times the wall concentration divided by the consumption. With a diffusion coefficient of 2.0e-9 square metres per second, a solubility of 1.4e-3 moles per cubic metre per millimetre of mercury and a partial pressure of forty, the wall concentration is 0.056000 moles per cubic metre. Resting skeletal muscle, consuming 0.3 millilitres per hundred grams per minute, gives 2.232143e-3 moles per cubic metre per second and a reach of 316.8 micrometres; moderate work gives 100.2 micrometres; and maximal exercise, fifty times the resting demand, gives 1.116071e-1 and a reach of 44.8 micrometres. Sixth, that length is not set by the resting demand. The observed capillary spacing, the Krogh radius, is twenty to eighty micrometres. At the resting demand the reach extends to 316.8 micrometres, four times further, so the observed spacing looks excessive; what it matches is the demand at maximal exercise. The length is set by the maximum rather than the average. Since the reach falls as the inverse square root of the consumption, a demand fifty times larger shrinks the distance by the square root of fifty, 7.0711, and 316.8 divided by 7.0711 is 44.8, so the first and third rows correspond exactly. Stated honestly, this paper reports an agreement and no more; it does not claim that the arrangement of capillaries is set by the maximal demand, since that would require evidence from the developmental side which this paper does not have, and treating the tissue as isotropic and homogeneous is also a departure from the real thing. Closing. The four pi that Paper 02 found is not something that appears everywhere. It appears only when point source, three dimensions and isotropy hold together, and raising the dimension of the source by one alone removes it. What appears after it has gone is two pi and a logarithm, and the fall points the other way: over the same distance the point source becomes 25.0 times weaker while the line source becomes 5.64 times stronger. And sources in living tissue are often not points. A capillary is a cylinder, and the reach around it is 316.8 micrometres at rest and 44.8 at maximal exercise, the latter being what matches the observed spacing. One thing separates them, which is writing down the dimension of the source. Write it down, and the occasions where four pi may be used separate from the occasions where using it is wrong by a factor of two. Do not write it down, and one fits a one-over-r to a field that grows stronger with distance. On the making of this work: The ideas and content of this work stem from the author's own considerations. Assistance from an AI (a large language model) was used for structuring, English translation, and checking the algebra. Any remaining errors or misinterpretations are solely the author's. Feedback and corrections are sincerely appreciated. ----- 論文02 は、逆二乗場に純粋な立体角 4π が現れることを示した。本稿が問うのは、その 4π はいつ現れないのかである。答は、源が点でなくなった瞬間である。源の次元を一つ上げるだけで、4π は 2π になり、1/r という落ち方は対数に変わる。新しい数学定理も新しい法則も主張しない。 本稿の射程(射程注記):新しい数学定理も新しい法則も主張しない。点源・線源・面源の電位、容積導体という近似、クローの円柱模型は、いずれも標準的である。電気生理学を作らない──使うのは三つの解と、一つの平方根と、単位の換算だけである。4π の由来を論じない──論文02 が扱う。本稿は逆に、その 4π が消える条件だけを扱う。逆問題に入らない──論文99 は境界データから内部の導電率を復元できるかを問い、論文159 は不良設定を分けた。本稿は順方向であり、源の形が与えられたときにどの立体角が出るかだけを見る。捕捉率を扱わない──論文249 はスモルコフスキーの捕捉率を球・定常・吸収境界で扱った。本稿は円柱で、消費項があり、問うのが捕捉率ではなく到達距離である。生理学を主張しない──毛細血管の配置がなぜそうなっているかの説明を与えない。一致を報告するだけであり、因果を述べない。生体組織を等方均質として扱っている──現実の組織は異方的で不均質である。数値は桁を見るための値であり、特定の組織の測定値ではない。源を無限に長い円柱として扱っている──有限長では端で 4π 側へ戻る。本稿が言うのは、無限に長い極限で 4π が出ないという一点である。 既刊との関係。論文02 は逆二乗場に純粋立体角 4π が現れることを示した。本稿はその 4π の定義域を書く。点源・三次元・等方の三つが揃ったときだけである。論文249 は拡散が捕らえる 4π と場が広がる 4π が同根だと示した。本稿は同じ 4π が出ない場合を並べる。同根の確認と、定義域の確認は対になっている。論文01 は n = 2 が唯一であることを四方向から示した。本稿が動かすのは指数ではなく源の次元であり、別の軸である。論文199 は帰ってこられるかを分母の指数が決めると書いた。本稿の第4節も分母の形を並べるが、問うのは再帰性ではなく落ち方である。論文112 は分解能を六つの別根の一つに数えた。本稿は測定の分解能を扱わない。 第一に、点源では 4π がそのまま分母に立つ。電流 1 マイクロアンペア、導電率 0.33 S/m で、電位は 5 mm で 48.2288 マイクロボルト、10 mm で 24.1144、20 mm で 12.0572、50 mm で 4.8229 になる。距離を 2 倍にすると電位はちょうど半分になり、分母の 4π=12.566371 は球の全立体角そのものである。 第二に、これが本稿の芯である。源を無限に長い線にすると 4π が消え、2π と対数が出る。単位長あたり 10 のマイナス4乗アンペアなら前係数は 4.822877e-5 ボルトで、電位差はその前係数に距離の比の対数を掛けたものになる。比が 2 なら 33.4296 マイクロボルト、10 なら 111.0508、100 なら 222.1017 である。源の次元を 0 から 1 に上げただけで、分母の立体角が 12.566371 から 6.283185 に半分になった。 第三に、落ち方そのものが逆を向く。線源の基準を 1 mm にとって同じ距離で並べると、点源は 2 mm で 120.5719 マイクロボルト、50 mm で 4.8229 まで 25.0 分の 1 に落ちるのに、線源は 33.4296 から 188.6721 へ 5.64 倍に増える。同じ距離を離れているのに向きが逆である。源の形を書かずに「距離が離れれば弱くなる」と言うことはできない。分離子は源の次元を一つ上げることだけであって、媒質も、電流も、空間の次元数も動かしていない。 第四に、三つの源で立体角を点呼する。次元 0 の点は 4π=12.566371 で 1/r に落ち、次元 1 の線(円柱)は 2π=6.283185 で対数に落ち、次元 2 の面(層)は 2π で距離に依らない。源の次元を一つ上げるたびに、落ち方が一段平らになる。4π が出るのは一行目だけであり、条件を書き出すと点源・三次元・