Chapter XI

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A TESSARACTIC FIGURE AND ITS PROJECTIONS.

We will now consider a fourth-dimensional shape composed of tessaracts, and the manner in which we can obtain a conception of it. The operation is precisely analogous to that described in chapter VI., by which a plane being could obtain a conception of solid shapes. It is only a little more difficult in that we have to deal with one dimension or direction more, and can only do so symbolically.

We will assume the shape to consist of a certain number of the 81 tessaracts, whose names we have given on p. 168. Let it consist of the thirteen tessaracts: Urna, Moles, Plebs, Frenum, Pallor, Tessera, Cudo, Vitta, Cura, Penates, Polus, Orcus, Lacerta.

Firstly, we will consider what appearances or projections these tessaracts will present to us according as the tessaractic set touches our space with its (_a_) Mala cubes, (_b_) Vesper cubes, (_c_) Pluvium cubes, or (_d_) Lar cubes. Secondly, we will treat the converse question, how the shape can be determined when the projections in each of those views are given.

Let us build up in cubes the four different arrangements of the tessaracts according as they enter our space on their Mala, Vesper, Pluvium or Lar sides. They can only be printed by symbolizing two of the directions. In the following tabulations the directions Y, X will at once be understood. The direction Z (expressed by the wavy line) indicates that the floors of nine, each printed nearer the top of the page, lie above those printed nearer the bottom of it. The direction W is indicated by the dotted line, which shows that the floors of nine lying to the left or right are in the W direction (Ana) or the -W direction (Kata) from those which lie to the right or left. For instance, in the arrangement of the tessaracts, as Malas (Table A) the tessaract Tessara, which is exactly in the middle of the eighty-one tessaracts has

Domitor on its right side or in the X direction.
Ocrea on its left „ „ -X „
Glans away from us „ „ Y „
Cudo nearer to us „ „ -Y „
Sacerdos above it „ „ Z „
Cura below it „ „ -Z „
Lacerta in the Ana or W „
Pallor in the Kata or -W „

Similarly Cervix lies in the Ana or W direction from Urna, with Thyrsus between them. And to take one more instance, a journey from Saltus to Arcus would be made by travelling Y to Remus, thence -X to Sector, thence Z to Mars, and finally W to Arcus. A line from Saltus to Arcus is therefore a diagonal of the set of 81 tessaracts, because the full length of its side has been traversed in each of the four directions to reach one from the other, _i.e._ Saltus to Remus, Remus to Sector, Sector to Mars, Mars to Arcus.

TABLE A.

Mala presentation of 81 Tessaracts.

Block A:
Arcus Ovis Portio
Laurus Tigris Segmen
Axis Troja Aries

Block B:
Ara Vomer Pluma
Praeda Sacerdos Hydra
Cortex Mica Flagellum

Block C:
Mars Merces Tyro
Spicula Mora Oliva
Comes Tibicen Vestis

Block D:
Postis Clipeus Tabula
_Orcus_ _Lacerta_ Testudo
Verbum Luctus Anguis

Block E:
Pilum Glans Coins
Ocrea _Tessera_ Domitor
Cardo _Cudo_ Malleus

Block F:
Ala Cortis Aer
Uncus‡ _Pallor_‡ Tergum
Ostrum Bidens‡ Scena

Block G:
Telum Nepos Angusta
_Polus_ _Penates_ Vulcan
Cervix Securis Vinculum

Block H:
Agmen Lacus Arvus
Crates _Cura_ Limen
Thyrsus _Vitta_ Sceptrum

Block I:
Sector Hama Remus
_Frenum_‡ _Plebs_‡ Sypho
_Urna_‡ _Moles_‡ Saltus

TABLE B.

Vesper presentation of 81 Tessaracts.

Block A:
Portio Pluma Tyro
Segmen Hydra Oliva
Aries Flagellum Vestis

Block B:
Ovis Vomer Merces
Tigris Sacerdos Mora
Troja Mica Tibicen

Block C:
Arcus Ara Mars
Laurus Praeda Spicula
Axis Cortex Comes

Block D:
Tabula Colus Aer
Testudo Domitor Tergum
Anguis Malleus Scena

Block E:
Clipeus Glans Cortis
_Lacerta_* _Tessera_* _Pallor_*
Luctus* _Cudo_* Bidens*

Block F:
Postis Pilum Ala
_Orcus_* Ocrea* Uncus*
Verbum† Cardo† Ostrum†

Block G:
Angusta Arvus Remus
Vulcan Limen Sypho
Vinculum Sceptrum Saltus

Block H:
Nepos Lacus Hama
_Penates_* _Cura_* _Plebs_*
Securis* _Vitta_* _Moles_*

Block I:
Telum Agmen Sector
_Polus_* Crates* _Frenum_*
Cervix* Thyrsus* _Urna_*

TABLE C.

Pluvium presentation of 81 Tessaracts.

Block A:
Mars Merces Tyro
Ara Vomer Pluma
Arcus Ovis Portio

Block B:
Spicula Mora Oliva
Praeda Sacerdos Hydra
Laurus Tigris Segmen

Block C:
Comes Tibicen Vestis
Cortex Mica Flagellum
Axis Troja Aries

Block D:
Ala Cortis Aer
Pilum Glans Colus
Postis Clipeus Tabula

Block E:
Uncus* _Pallor_* Tergum
Ocrea* _Tessera_* Domitor
_Orcus_* _Lacerta_* Testudo

Block F:
Ostrum† Bidens† Scena
Cardo† _Cudo_* Malleus
Verbum† Luctus† Anguis

Block G:
Sector Hama Remus
Agmen Lacus Arvus
Telum Nepos Angusta

Block H:
_Frenum_* _Plebs_* Sypho
Crates* _Cura_* Limen
_Polus_* _Penates_* Vulcan

Block I:
_Urna_* _Moles_* Saltus
Thyrsus* _Vitta_* Sceptrum
Cervix† Securis† Vinculum

TABLE D.

Lar presentation of 81 Tessaracts.

Block A:
Mars Merces Tyro
Spicula Mora Oliva
Comes Tibicen Vestis

Block B:
Ala Cortis Aer
Uncus _Pallor_* Tergum
Ostrum Bidens Scena

Block C:
Sector Hama Remus
_Frenum_* _Plebs_* Sypho
_Urna_* _Moles_* Saltus

Block D:
Ara Vomer Pluma
Proeda Sacerdos Hydra
Cortex Mica Flagellum

Block E:
Pilum Glans Colus
Ocrea _Tessera_* Domitor
Cardo _Cudo_* Malleus

Block F:
Agmen Laurus Arvus
Crates _Cura_* Limen
Thyrsus _Vitta_* Sceptrum

Block G:
Arcus Ovis Portio
Laurus Tigris Segmen
Axis Troja Aries

Block H:
Postis Clipeus Tabula
_Orcus_* _Lacerta_* Testudo
Verbum Luctus Anguis

Block I:
Telum Nepos Angusta
_Polus_* _Penates_* Vulcan
Cervix Securis Vinculum

The relation between the four different arrangements shown in the tables A, B, C, and D, will be understood from what has been said in chapter VIII. about a small set of sixteen tessaracts. A glance at the lines, which indicate the directions in each, will show the changes effected by turning the tessaracts from the Mala presentation.

In the Vesper presentation:

The tessaracts--
(_e.g._ Urna, Ostrum, Comes), which ran Z still run Z.
(_e.g._ Urna, Frenum, Sector), „ Y „ Y.
(_e.g._ Urna, Moles, Saltus), „ X now run W.
(_e.g._ Urna, Thyrsus, Cervix), „ W „ -X.

In the Pluvium presentation:

The tessaracts--
(_e.g._ Urna, Ostrum, Comes), which ran Z still run Z.
(_e.g._ Urna, Moles, Saltus), „ X „ X.
(_e.g._ Urna, Frenum, Sector), „ Y now run W.
(_e.g._ Urna, Thyrsus, Cervix), „ W „ -Y.

In the Lar presentation:

The tessaracts--
(_e.g._ Urna, Moles, Saltus), which ran X still run X.
(_e.g._ Urna, Frenum, Sector), „ Y „ Y.
(_e.g._ Urna, Ostrum, Comes), „ Z now run W.
(_e.g._ Urna, Thyrsus, Cervix), „ W „ -Z.

This relation was already treated in chapter IX., but it is well to have it very clear for our present purpose. For it is the apparent change of the relative positions of the tessaracts in each presentation, which enables us to determine any body of them.

In considering the projections, we always suppose ourselves to be situated Ana or W towards the tessaracts, and any movement to be Kata or -W through our space. For instance, in the Mala presentation we have first in our space the Malas of that block of tessaracts, which is the last in the -W direction. Thus, the Mala projection of any given tessaract of the set is that Mala in the extreme -W block, whose place its (the given tessaract’s) Mala would occupy, if the tessaractic set moved Kata until the given tessaract reached our space. Or, in other words, if all the tessaracts were transparent except those which constitute the body under consideration, and if a light shone through Four-space from the Ana (W) side to the Kata (-W) side, there would be darkness in each of those Malas, which would be occupied by the Mala of any opaque tessaract, if the tessaractic set moved Kata.

Let us look at the set of 81 tessaracts we have built up in the Mala arrangements, and trace the projections in the extreme -W block of the thirteen of our shape. The latter are printed in italics in Table A, and their projections are marked ‡.

Thus the cube Uncus Mala is the projection of the tessaract Orcus, Pallor Mala of Pallor and Tessera and Tacerta, Bidens Mala of Cudo, Frenum Mala of Frenum and Polus, Plebs Mala of Plebs and Cura and Penates, Moles Mala of Moles and Vitta, Urna Mala of Urna.

Similarly, we can trace the Vesper projections (Table B). Orcus Vesper is the projection of the tessaracts Orcus and Lacerta, Ocrea Vesper of Tessera, Uncus Vesper of Pallor, Cardo Vesper of Cudo, Polus Vesper of Polus and Penates, Crates Vesper of Cura, Frenum Vesper of Frenum and Plebs, Urna Vesper of Urna and Moles, Thyrsus Vesper of Vitta. Next in the Pluvium presentation (Table C) we find that Bidens Pluvium is the projection of the tessaract Pallor, Cudo Pluvium of Cudo and Tessera, Luctus Pluvium of Lacerta, Verbum Pluvium of Orcus, Urna Pluvium of Urna and Frenum, Moles Pluvium of Moles and Plebs, Vitta Pluvium of Vitta and Cura, Securis Pluvium of Penates, Cervix Pluvium of Polus. Lastly, in the Lar presentation (Table D) we observe that Frenum Lar is the projection of Frenum, Plebs Lar of Plebs and Pallor, Moles Lar of Moles, Urna Lar of Urna, Cura Lar of Cura and Tessara, Vitta Lar of Vitta and Cudo, Penates Lar of Penates and Lacerta, Polur Lar of Polus and Orcus.

Secondly, we will treat the converse problem, how to determine the shape when the projections in each presentation are given. Looking back at the list just given above, let us write down in each presentation the projections only.

Mala projections:

Uncus, Pallor, Bidens, Frenum, Plebs, Moles, Urna.

Vesper projections:

Orcus, Ocrea, Uncus, Cardo, Polus, Crates, Frenum, Urna, Thyrsus.

Pluvium projections:

Bidens, Cudo, Luctus, Verbum, Urna, Moles, Vitta, Securis, Cervix.

Lar projections:

Frenum, Plebs, Moles, Urna, Cura, Vitta, Polus, Penates.

Now let us determine the shape indicated by these projections. In now using the same tables we must not notice the italics, as the shape is supposed to be unknown. It is assumed that the reader is building the problem in cubes. From the Mala projections we might infer the presence of all or any of the tessaracts written in the brackets in the following list of the Mala presentation.

(Uncus, Ocrea, Orcus); (Pallor, Tessera, Lacerta);

(Bidens, Cudo, Luctus); (Frenum, Crates, Polus);

(Plebs, Cura, Penates); (Moles, Vitta, Securis);

(Urna, Thyrsus, Cervix).

Let us suppose them all to be present in our shape, and observe what their appearance would be in the Vesper presentation. We mark them all with an asterisk in Table B. In addition to those already marked we must mark (†) Verbum, Cardo, Ostrum, and then we see all the Vesper projections, which would be formed by all the tessaracts possible from the Mala projections. Let us compare these Vesper projections, viz. Orcus, Ocrea, Uncus, Verbum, Cardo, Ostrum, Polus, Crates, Frenum, Cervix, Thyrsus, Urna, with the given Vesper projections. We see at once that Verbum, Ostrum, and Cervix are absent. Therefore, we may conclude that all the tessaracts, which would be implied as possible by their presence, are absent, and of the Mala possibilities may exclude the tessaracts Bidens, Luctus, Securis, and Cervix itself. Thus, of the 21 tessaracts possible in the Mala view, there remain only 17 possible, both in the Mala and Vesper views, viz. Uncus, Ocrea, Orcus, Pallor, Tessera, Lacerta, Cudo, Frenum, Crates, Polus, Plebs, Cura, Penates, Moles, Vitta, Urna, Thyrsus. This we call the Mala-Vesper solution.

Next let us take the Pluvium presentation. We again mark with an asterisk in Table C the possibilities inferred from the Mala-Vesper solution, and take the projections those possibilities would produce. The additional projections are again marked (†). There are twelve Pluvium projections altogether, viz. Bidens, Ostrum, Cudo, Cardo, Luctus, Verbum, Urna, Moles, Vitta, Thyrsus, Securis, Cervix. Again we compare these with the given Pluvium projections, and find three are absent, viz. Ostrum, Cardo, Thyrsus. Hence the tessaracts implied by Ostrum and Cardo and Thyrsus cannot be in our shape, viz. Uncus, Ocrea, Crates, nor Thyrsus itself. Excluding these four from the seventeen possibilities of the Mala-Vesper solution we have left the thirteen tessaracts: Orcus, Pallor, Tessera, Lacerta, Cudo, Frenum, Polus, Plebs, Cura, Penates, Moles, Vitta, Urna. This we call the Mala-Vesper-Pluvium solution.

Lastly, we have to consider whether these thirteen tessaracts are consistent with the given Lar projections. We mark them again on Table D with an asterisk, and we find that the projections are exactly those given, viz. Frenum, Plebs, Moles, Urna, Cura, Vitta, Polus, Penates. Therefore, we have not to exclude any of the thirteen, and can infer that they constitute the shape, which produces the four different given views or projections.

In fine, any shape in space consists of the possibilities common to the projections of its parts upon the boundaries of that space, whatever be the number of its dimensions. Hence the simple rule for the determination of the shape would be to write down all the possibilities of the sets of projections, and then cancel all those possibilities which are not common to all. But the process adopted above is much preferable, as through it we may realize the gradual delimitation of the shape view by view. For once more we must remind ourselves that our great object is, not to arrive at results by symbolical operations, but to realize those results piece by piece through realized processes.

APPENDICES.

APPENDIX A.

This set of 100 names is useful for studying Plane Space, and forms a square 10 × 10.

Aiōn Bios Hupar Neas Kairos Enos Thlipsis Cheimas Theion Epei

Itea Hagios Phaino Geras Tholos Ergon Pachūs Kiōn Eris Cleos

Loma Etēs Trochos Klazo Lutron Hēdūs Ischūs Paigma Hedna Demas

Numphe Bathus Pauo Euthu Holos Para Thuos Karē Pylē Spareis

Ania Eōn Seranx Mesoi Dramo Thallos Aktē Ozo Onos Magos

Notos Mēnis Lampas Ornis Thama Eni Pholis Mala Strizo Rudon

Labo Helor Rupa Rabdos Doru Epos Theos Idris Ēdē Hepo

Sophos Ichor Kaneōn Ephthra Oxis Lukē Blue Helos Peri Thelus

Eunis Limos Keedo Igde Matē Lukos Pteris Holmos Oulo Dokos

Aeido Ias Assa Muzo Hippeus Eōs Atē Akme Ōrē Gua

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A New Era of ThoughtChapter XI

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