[Home]
An Introduction to The Geometry of N Dimensions
|
|
CONTENTS (-8 to -6)
|
|
PREFACE (-5 to -1)It is scarcely necessary to apologise for writing a book on n-dimensional geometry. One should regret rather the comparative neglect which the subject has suffered at the hands of British mathematicians.* Yet one may almost say that this country was its home of origin, for, with the exception of a few previous sporadic references, the first paper dealing explicitly with geometry of n dimensions was one by Cayley in 1843, and the importance of the subject was recognised from the first by three of our most famous pure mathematicians — Cayley, Clifford, and Sylvester. On the Continent the classical works of Grassmann and Schlafli attracted at first no attention. Schlafli’s remarkable memoir, in fact, failed to secure publication, and in spite of Cayley’s gallant attempt at rescue by translating and publishing part of it in the “ Quarterly Journal” it remained unknown until it was found and published several years after the author’s death, and fifty years after it was written. By that time Schlegel and others in Germany had made the subject well known, but mostly in its metrical aspect. The wonderful projective geometry of hyperspace has been almost entirely the product of the gifted Italian school of geometers ; though this branch also was inaugurated by a British mathematician, W. K. Clifford, in 1878.
The present introduction deals with the metrical and to a slighter extent with the projective aspect. A third aspect, which has attracted much attention recently, from its application to relativity, is the differential aspect. This is altogether excluded from the present book. In writing this book I have not attempted to produce a complete systematic treatise, but have rather selected certain representative topics which not only illustrate the extensions of theorems of three-dimensional geometry, but reveal results which are unexpected and where analogy would be a faithless guide. The first four chapters explain the fundamental ideas of incidence, parallelism, perpendicularity, and angles between linear spaces ; and in Chapter I there is an excursus into enumerative geometry which may be omitted on a first reading. Chapters V and VI are analytical, the former projective, the latter largely metrical. In the former are given some of the simplest ideas relating to algebraic varieties, and a more detailed account of quadrics, especially with reference to their linear spaces. In the latter there are given, in addition to the ordinary cartesian formulae, some account and applications of the Plucker-Grassmann coordinates of a linear space, and applications to line-geometry. The remaining chapters deal with polytopes, and contain, especially in Chapter IX, some of the elementary ideas in analysis situs. Chapter VIII treats of the content of hyperspatial figures, and the final chapter establishes the regular polytopes. A number of references have been given at the ends of the chapters. Some of these are the original works in which the various theories were first expounded, others are a selection of more recent works in which a fuller account may be found. Reference may be made to the author’s “ Bibliography of Non-Euclidean Geometry, including the theory of parallels, the foundations of geometry, and space of n dimensions” (London : Harrison, for the University of St. Andrews. 1911), which contains, in addition to a chronological catalogue, a detailed subject index and an index of authors. I am indebted in particular to Schoute’s “ Mehr-dimensionale Geometric” (Leipzig, 2 vols., 1902 and 1905), Bertini’s “ Introduzione alia geometria proiettiva degli iperspazi” (Pisa, 1907), and the various articles of the “ Encyklopadie der mathematischen Wissen-schaften.” For assistance in correcting the proofs I have to thank Mr. F. F. Miles, M.A., Lecturer in Mathematics at this College. D. M. Y. S. Victoria University College, Wellington, N.Z., May , 1929. |
|
REFERENCES
|
|
Index (193-196)
People Index
|
|
CHAPTER I: FUNDAMENTAL IDEAS1. Origins of Geometry. Geometry for the individual begins intuitional ly and develops by a co-ordination of the senses of sight and touch. Its history followed a similar course. The crude ideas of shape, bulk, superficial extent, and length became analysed, refined, and made abstract, and led to the conception of geometrical figures. The development started with the solid ; surface and line, without solidity, were later abstractions. Witness the inability of most animals and some primitive races of men to recognise a picture. Having no depth, except such as is imitated by the skilfulness of the drawing or shading, it conveys to the undeveloped intelligence only an impression of flat regions of contrasted colouring. When the power of abstraction had proceeded to the extent of conceiving surfaces apart from solids, plane geometry arose. The idea of dimensionality was then formed, when a region of two dimensions was recognised within the three-dimensional universe. This stage had been reached when Greek geometry started. It was many centuries, however, before the human mind began to conceive of an upward extension to the idea of dimensionality, and even now this conception is confined to the comparatively very small class of mathematicians and philosophers. 2. Extension of the Dimensional Idea. There are two main ways in which we may arrive at an idea of higher dimensions : one geometrical, by extending in the upward direction the series of geometrical elements, point, line, surface, solid ; the other by invoking algebra and giving extended geometrical interpretations to algebraic relationships. In whatever way we may proceed we are led to the invention of new [1] 2 GEOMETRY OF N DIMENSIONS [i. 3 elements which have to be defined strictly and logically if exact deductions are to be made. A great deal is suggested by analogy, but while analogy is often a useful guide and stimulus, it provides no proofs, and may often lead one astray if not supplemented by logical reasoning. If we follow the geometrical method the only safe course is that which was systematically laid down for the first time by Euclid, that is to lay down a basis of axioms or assumptions. When we leave the field of sensuous perception and can no longer depend upon intuition as a guide, our axioms will no longer be self-evident truths,'* but simply statements, assumed without proof, as a basis for future deductions. 3. Definitions and Axioms. In geometry there are objects which have to be defined, and relationships between these objects which have to be deduced either from the definitions or from other simpler relationships. In defining an object we must make reference to some simpler object, hence there must be some objects which have to be left undefined, the indefinables. Similarly, in deducing relations from simpler ones we must arrive back at certain statements which cannot be deduced from anything simpler ; these are the axioms or unproved propositions. The whole science of geometry can thus be made to rest upon a set of definitions and axioms. The actual choice of fundamental definitions and axioms is to a certain extent arbitrary, but there are certain principles which have to be considered in making a choice of axioms. These are : — (1) Self-consistency, The set of axioms must be logically self-consistent. No axiom must be in conflict with deductions from any of the other axioms. (2) Non-redundance or Independence, This condition is not a necessary one, but in a logical scheme it is desirable. Pedagogically the condition is frequently ignored. ( 3 ) Categoricalness. This means not only that the set of axioms should be complete and sufficient for the development of the science, but that it should be possible to construct only a unique set of entities for which the axioms are valid. It is doubtful whether any set of axioms can be strictly categorical. If any set of entities is constructed so as to satisfy the axioms, FUNDAMENTAL IDEAS [3] L 4] it is nearly always, if not always, possible to change the ideas and construct another set of entities also satisfying the axioms. Thus with the ordinary ideas of point and straight line in plane geometry the axioms can still be applied when instead of a point we substitute a pair of numbers (or, j), and instead of straight line an equation of the first degree in Jtr and jy ; corresponding to the incidence of a point with a straight line we have the fact that the values of x and jy satisfy the equation. It is desirable, in fact, that the set of axioms should not be categorical, for thereby they are given a wider field of validity, and propositions proved for the one set of entities can be transferred at once to another set, perhaps in a different branch of mathematics. 4. The Axioms of Incidence. As indefinables we shall choose first the pointy straight limy and plane. With regard to these we shall proceed to make certain statements, the axioms. If these should appear to be very obvious, and as if they might be taken for granted, it will be a good corrective for the reader to replace the words point and straight line, which he must remember are not yet defined, by the names of other objects to which the axioms may be made to apply, such as “ committee member” and '‘committee.” Following Hilbert we divide the axioms into groups. THE AXIOMS Group I AXIOMS OF INCIDENCE OR CONNECTION I. I. Any two distinct points uniquely determine a straight line. We imagine a collection of individuals who have a craze for organisation and form themselves into committees. The committees are so arranged that every person is on a committee along with each of the others, but no two individuals are to be found together on more than one committee. I. 2. If A, B are distinct points there is at least one point not on the straight line AB. This is an “ existence-postulate.” I. 3. Any three non-collinear points determine a plane. [4] GEOMETRY OF N DIMENSIONS [i. 5 I. 4. If two distinct points B both belong to a plane a, every point of the straight line AB belongs to a. From I. I it follows that two distinct straight lines have either one or no point in common. From I. 4 it follows that a straight line and a plane have either no point or one point in common, or else the straight line lies entirely in the plane ; from 1. 3 and 4 two distinct planes have either no point, one point, or a whole straight line in common. I. 5. If Ay B, C are non-collinear points there is at least one point not on the plane ABC. I. 2 and 5 are existence-postulates ; 2 implies two-dimensional geometry, and 5 three-dimensional. The next of Hilbert’s axioms is that if two planes have one point A in common they have a second point B in common, and therefore by I. 4 they have the whole straight line AB in common. If this is assumed it limits space to three dimensions. 5. Projective Geometry. There is a difficulty in determining all the elements of space by means of the existence postulates and other axioms, for while Axiom I. i postulates that any two points determine a line, there is no axiom which secures that any two lines in a plane will determine a point. In fact, in euclidean geometry this is not true since parallel lines have no point in common. For the present therefore we shall confine ourselves to a simpler and more symmetrical type of geometry, projective geometry ^ for which we add the following axiom : L i'. Any two distinct straight lines in a plane uniquely determine a point. 6. Construction of Three-dimensional Space. We may proceed now to obtain all the elements, points, lines, etc., of space with the help of the existence postulates and other axioms. Starting with two points A, B we determine the line AB, and on this we shall suppose all the points determined. Taking a third point C, not on AB, a plane ABC is determined. In this plane there are determined : first, the three points A, B, C, then the three lines BC, CA, AB and all their points ; then all lines determined by two points one on each of two of these lines, and since every line in the plane meets these three lines all the lines of the plane are thus determined ; finally any point [5] FUNDAMENTAL IDEAS I- 7] of the plane is determined by the intersection of two lines which have already been determined, e.g. if P is any point of the plane, the line PA cuts BC in a point L and PB cuts CA in a point M ; there are therefore two points, L on BC and M on CA, such that LA and MB determine P. We next take a point D not in the plane ABC, and determine planes, lines and points as follows : first, the planes DBC, DCA, DAB and all the lines and points in these planes ; then the lines determined by joining D to the points of ABC, and all the points on these lines, and all the lines and planes determined by any two or any three of the points thus determined. Jf now P is any point we cannot be sure that it is one of the points thus determined unless the line DP meets the plane ABC. P'or this we could assume as an axiom : every line meets every plane in a point,’' which is true for projective geometry of three dimensions ; but a weaker axiom, which is true also in euclidean geometry, is sufficient, viz. Hilbert’s axiom : I. 6. If two planes have a point A in commo 7 t^ they have a second point B in common. Then if P is any point, the planes PAD and ABC, which have the point A common, have another point, say Q, common, and the lines AQ and DP which lie in the same plane determine a point R which lies in the line DP and also in the plane ABC. Hence DP meets the plane ABC, and therefore all points are obtained by this process. If / is any line (Fig. i), determined by the two points P, Q, the planes PQD and ABD, having D in common, intersect in a line which cuts PQ in N, say, and AB in E ; then PQ and CE lie in the same plane and therefore meet in a point Z. Hence every line cuts the plane ABC. 7. If a is any plane (Fig. 2), determined by the three points P, Q, R, and O is a point not in this plane, the lines OP, OQ, OR cut ABC in P', Q', R'. QR and Q'R', being in the same plane, intersect in a point X. Similarly RP and R'P' intersect in Y, and PQ and P'Q' in Z. Hence every plane cuts the plane ABC. The points X, Y, Z all lie in both of the planes PQR and P'Q'R' and are therefore collinear. This theorem is known as DesargueP Theoremy PQR and P'Q'R' being two triangles in [6]GEOMETRY OF N DIMENSIONS [l. 8 perspective. (It is interesting to note in passing that the corresponding theorem for two triangles in perspective in the same plane cannot be proved without the assumption of three dimensions or some special axiom in addition to those which we have assumed) We can now prove that any two planes intersect in a line. Let a and a' be any two planes. We have seen that every plane cuts ABC in a straight line. Let a and a cut ABC in the lines / and I, These lines intersect in a point P which is therefore common to the two planes, and therefore the two planes, having a point in common, intersect in a straight line. Next, every line cuts every plane in a point. For if is any plane through the given line /, this plane cuts the given plane a in a line m, and the two lines I and lying in the same plane, intersect in a point, which is common to I and a. Lastly, any three planes have a point in common. For the planes a, )3 intersect in a line /, and / cuts y in a point ; this point is common to the three planes. 8. Extension to Four Dimensions. All the points, lines and planes determined from four given non-coplanar points form a three-dimensional region, which is ordinary projective space of three dimensions. We proceed to postulate the existence of at least one point E not in this region. The three-dimensional region is not now the whole of space, but I. 9] FUNDAMENTAL IDEAS 7 will be called a hyperplane lying in hyperspace. A hyperplane is thus determined by four points. We may determine similarly the hyperplanes ABCE, ABDE, etc., and the planes, lines and points in them, and further the hyperplanes, planes, and lines determined by four, three, or two points not all lying in one of these hyperplanes. The hyperplanes ABCE and ABDE have the three points A, B, E in common and therefore the plane ABE. We may show that any two hyperplanes in this region have a plane in common. For example, if a and a are two hyperplanes determined by four points P, Q, R, S and P'l Q', S' lying on EA, EB, EC, ED respectively, PQ and P'Q', being in the plane EAB, cut in a point Z, similarly QR and Q'R' cut in a point X, and RP and RT' in a point Y ; PS and P'S' cut in U, QS and Q'S' in V, RS and R'S' in W. The six points X, Y, Z, U, V, W are thus all common to a and a'. These six points form the vertices of a complete quadrilateral (Fig. 3) whose sides are XYZ, the intersection of the planes PQR and P'Q'R', XVW, the intersection of QRS and Q'R'S', etc. The six points are thus coplanar, and this plane is common to the two hyperplanes a and a'. Thus in this region, space of four dimensions, two hyperplanes intersect in a plane. Similarly three hyperplanes intersect in a line, four hyperplanes in a point, while five hyperplanes do not in general have any point in common. A hyperplane cuts a plane (intersection of two hyperplanes) in a line, and a line in a point. Two planes, each the intersection of two hyperplanes, have in general only one point in common, a plane and a line in general have no point in common. A hyperplane is determined by four points, a point and a plane, or by two skew lines. 9. Degrees of Freedom : Dimensions. A point in a line is said to have one degree of freedom, in a plane two, in a hyperplane three, and in the hyperspacial region four. The point being taken as the element, a line is said to be of one TY 8 GEOMETRY OF N DIMENSIONS [i. lo dimension, a plane two, a hyperplane three, and the hyperspacial region four. For a point to be in a given hyperplane one condition is required, or one degree of constraint, thus reducing the number of degrees of freedom from four to three. For a point to lie in a plane two conditions are required ; if it is to lie simultaneously in two planes four conditions are required and it is completely determined.
10. Extension to n Dimensions. We may now extend these ideas straightaway
to n dimensions, and at the same time acquire both greater generality
and greater succinctness in expression. The series point, line, plane, hyperplane
(or as it is more explicitly termed three-jiat\ . . ., ;^-flat are regions
determined by one, two, three, four, . . ., + 1 points, and having zero,
one, two, three, . . ., « dimensions, i.e. an r-flat is determined hy
r + 1 points, and every /-flat (/
11. Independent Points. If / + 1 points uniquely determine a p-flat they
must not be contained in the same (p-1)-flat. Also no r of them (r /) must
be contained in the same {r - 2)-flat, for this {r - 2)-flat, which is determined
by r - 1 points, together with the remaining / + 1 - r points, would determine
a (/ - 1)-flat. We shall call a system of / -f i points, no r of which lie
in the same {r - 2)-flat, a system of linearly independent points. Any /
+ 1 points of a /-flat, if they are linearly independent, can be chosen to
determine the /-flat.
12. Consider a /-flat and a ^-flat, which are determined respectively by
/ + 1 and ^ + 1 points. If they have no point in common we have / + ^ + 2
independent points which determine a (/ + ^ -f i)-flat. Hence a p~jlat and
a q-flat taken arbitrarily lie in the same (/ + ^ + lyjlat. li p + q + 1
is greater than n the two flats will have a region in common. Let
this region be of dimensions r. In this region we may take r + 1 independent
points ; to determine the /-flat we require / - r additional points, and
to determine the ^-flat q - r additional points, i.e. altogether
I. 15]
FUNDAMENTAL IDEAS
9
(r + 1) + (/ - ^) + (5^ - r) = / + ^ - r + 1, and these determine R (p +
g - r)-flat. Hence
A p’jlat and a q-Jlat which have in common an r-jlat are both contained
in a {^p q - ryjlat ; if they have no point in common they are both contained
in a (p + q + 1 yflat.
A p-Jiat and a q-flat, which are both contained in a?i n-flat, have in common
{p + q - nfflat^ provided / + ^ ^ i.
If/ + q
This result may be proved otherwise, thus. Take any / + 1 fixed {n - /)-flats.
The /-flat cuts each of these in a point, and these / + 1 points determine
the /-flat. Each point has n - p degrees of freedom in its {n - /)-flat.
Hence the total number of degrees of freedom of the /-flat is (/ + 1)(/^
~ /).
If the /-flat has r points fixed, / -f i - r points are still required to
determine it, hence the number of degrees of freedom of the /-flat is {n
- /)(/ ~ r 4- i); hence also the number of degrees of freedom of a p-flat
lying in a given n-flat afid passing through a given r-flat is (n - /)(/
- r).
14. The degree of incidence of a /-flat and a ^-flat can be represented by
a fraction. Let / > ^. Complete incidence, when the /-flat contains the
^-flat, can be represented by i ; skewness, when they have no point in common,
by o. If they have in common an r-flat, the degree of incidence can be
represented by the fraction (r + 1) / (^^ + O*
15. Duality or Reciprocity. A /-flat and an (« - / ~ i)- flat in
Sn have the same constant-number (/ + 1 )(^ ~ /). A (i,
i) correspondence between points and {n - 1)-flats can be established in
various ways, so that to the line joining two points P, Q corresponds the
(n - 2)-flat of intersection of the two corresponding (n - 1)-flats. If three
points are collinear their
10
GEOMETRY OF N DIMENSIONS [i. i6
corresponding (n - 1)-flats pass through the same - 2 )-flat To a - 1)-flat,
which is determined by / given points, corresponds the (n - /)-flat common
to the (n - 1 )-flats which correspond to the p points.
16. Number of Conditions Required for a given Degree of Incidence. In S^
a /-flat has {n - /)(/ + 1) degrees of freedom, but if it passes through
a given r-flat it has only {n - /)(/ - r) degrees of freedom. Hence the number
of conditions that a p-flat in should pass through a given r~flat {n >/
> r) is {n - f){r + 1).
If the r-flat is free to move in a given ^-flat, it has {r + 1)(^ - 0 degrees
of freedom. Hence the nu^nber of conditions that a pfat and a q-flat in should,
intersect in an rfat is (r + 1)(n - p - q r). This implies that p + q
n + r. lip + q^n + r they intersect in a region of dimensions p q
- n, which is greater than r.
17. Incidence of a Linear Space with two or more Linear Spaces. Enumerative
Geometry. In S3 a line has 4 degrees of freedom (constant-number
4), and the number of conditions that it should intersect another line is
i. Hence a line which cuts a fixed line has 3 degrees of freedom, and the
whole system of lines all cutting a fixed line forms a three-dimensional
assemblage which is a particular case of a congruence ; the system of lines
cutting two fixed lines forms a two-dimensional assemblage, a particular
case of a complex ; and the system of lines cutting three fixed lines forms
a one-dimensional assemblage, a line-series. A line which is required to
cut four given lines is deprived of all freedom. The line is not, however,
uniquely determined. An important problem, which belongs to a branch of
mathematics called enumerative geometry^ is to determine the number of linear
spaces which satisfy given conditions, in number equal tp the constant-number
of the linear space. This problem can sometimes be solved directly by simple
geometrical considerations. As an example let us find the number of lines
in S5 (constant-number 8) which pass through a given point O (4
conditions) and cut two given planes a, b (2 conditions for each intersection).
The required line must lie in each of the 3-flats {Od) and (O^), hence it
is uniquely determined as the intersection of these two 3-flats.
I. 19]
FUNDAMENTAL IDEAS
II
18. Principle of Specialisation. The determination of the number of lines
which cut four lines in S3 is not so simple, and we apply a very
useful principle, called by Schubert the “conservation of number''
(Erhaltung der Anzahl), or “principle of specialisation." The principle
is that the number of elements determined will be the same if the determining
figures are specialised, provided the number does not thereby become infinite.
In simple cases it is equivalent to the statement in algebra that the number
of roots of an algebraic equation is not altered if the coefficients are
specialised in any way, unless, indeed, the equation becomes an identity.
As an example of the application of this principle let us complete the problem
to determine how many lines in Sg cut four given lines. Let the four lines
intersect in pairs : a and b in P, c and d in Q. A line which cuts both a
and b must either pass through P or lie in the plane {ab ^ ; and as P must
not lie in the plane {cd\ nor Q in the plane {ah), for in either of these
cases there would be an unlimited number of lines cutting all four lines,
the required line must either pass through both P and Q, or lie in both of
the planes {ab) and {cd). Hence there are two lines satisfying the given
conditions.
19. The general enumerative problem relating to linear spaces is : to find
the number of /-flats which have incidence of specified degrees with a given
set of linear spaces, the number of assigned conditions being equal to the
constant-number of the /-flat.
Let {n \ p, q \ r) represent the condition that a /-flat and a ^-flat in
should intersect in an r-flat. If P and Q represent any conditions, the product
PQ is taken to mean that both conditions must be simultaneously satisfied
; the sum P + Q that one or other of the conditions is satisfied. Thus (3
: 2, o : o)(3 : 2, I : i) means the condition that a plane in S3
should contain a given point and a given line; (3:1, 1:0/ means the condition
that a line in Sg should cut four given lines.
The number of simple conditions involved in (n :p, q : r) may be denoted
by C{n py q \ r), and we have proved that C{n :py q:r) ^ {r + 1){n - p --
q r).
The constant-number of a /-flat in Sn is equal to C(« :/,/ :/) - (/
+ 1){n - /).
12
GEOMETRY OF N DIMENSIONS [i. 20
We may also represent the number of elements determined by a given condition
by prefixing N to the symbol of the condition. Thus N(3 : I, i :o)^ = 2.
20. Duality in Enumerative Geometry. The duality between the /-flat and the
{n - p - 1)-flat extends to enumerative problems. Not only are the
constant-numbers equal, viz. (/ + 1){n ~ /), but we have also, as is easily
verified,
Q>{n :/,$': ^) == ^{n \ n - p - l, n ~ q - \ : n
- p - q -h r - l) ;
and the number of /-flats determined by (/ 4- i )(n - /) simple conditions
is equal to the number of (n - p - 1)-flats determined by the corresponding
reciprocal conditions. Thus in S4 there are two lines which lie
in a given 3-flat and cut four given planes, for the planes cut the 3-flat
in lines. The reciprocal statement is : there are two planes which pass through
a given point and cut four given lines ; i.e.
N(4 : I, 3 : i)(4 : i, 2 : o)^ - N(4 : 2, o : o)(4 : 2, i : o)l
21 . In two dimensions the only conditions are
C(2 : 0, 0 : 0) = 2 = C(2 : I, I : l),
C(2 : O, I : O) = I = C(2 ; I, o : o),
i.e. a point coincident with a given point, a line coincident with a given
line ; and a point incident with a given line, a line incident with a given
point. The constant-number K of a point or a line is 2, and we have only
two enumerative results with their duals, viz. : —
N(2 : o, 0 : o) = I, ie. one point is coincident with a given point,
N(2 : 0, I : o)^= I, ie, one point is incident with two given lines,
N(2 : I, 1 : i) = ly z,e. one line is coincident with a given line,
N(2 : I, 0 : o)^ == I, I.e. one line is incident withitwo given points.
The results may be exhibited more compactly in tabular form. For a specified
linear space let q^ denote that it cuts a given ^-flat in an r-flat. Then
for Sg all the results are represented in the following table. Point and
plane, being reciprocals, are grouped together.
I. 22 ]
FUNDAMENTAL IDEAS
C is the number of simple conditions, N is the number of elements determined.
In each row the sum of the numbers, each multiplied by the corresponding
value of C, is equal to the constant-number of the element K.
22. Incident Spaces in Four Dimensions. In general many of the combinations
are either trivial, or belong to a space of lower dimensions, e.g. in the
statement, that one line passes through a given point and intersects three
given lines, is the reciprocal of the statement that one plane lies in a
given 3 -flat and cuts three given lines (the plane is in fact determined
by the three points in which the given lines cut the 3 -flat).
The following is the complete table of results for S^, most of which the
reader will have no difficulty in verifying. Results marked “trivial”
are the fundamental determinations of an element or its reciprocal, e.g.
a plane determined by three points. “ S 3 ” indicates that the
result belongs to space of three dimensions.
... |
(Created August 30, 2024)