From at least the eighth century BCE, philosopher-scientists debated how to define place with respect to solid bodies. One question was whether two bodies could occupy the same place at the same time—and most philosophers agreed that such an interpenetration of bodies was impossible. Marvell refers to the debate about interpenetration in the poem “Fleckno, an English Priest at Rome,” when the Catholic priest Richard Flecknoe and his two visitors find it impossible to pass each other on the impoverished Flecknoe’s wretchedly narrow staircase. The narrator makes a metaphysical joke about the trinity: “‘There can no body pass / Except by penetration higher, where / Two make a crowd; nor can three persons here / Consist but in one substance’” (“Fleckno” 98-100).
Although I would not like to label Spenser’s poetry metaphysical, Marvell’s questions about the metaphysical nature of place—which appear throughout his work—draw deeply from Spenser. Both poets are intrigued by the placement of animate, inanimate, and geopolitical bodies. And both poets imagine physical impossibilities: bodies interpenetrating, a body meeting itself walking from another direction, self-intersecting bodies without a binary distinction between outer and inner surfaces, and a body occupying more than one place at once.
This paper will argue that both Spenser and Marvell thought of the scientific debates about bodies in space when they considered the place of nations. As both Spenser and Marvell knew, the agreement about the impossibility of interpenetration had made the definition of place all the more contentious. Hesiod and the atomists had believed that void space could exist apart from the objects that inhabited it and that place had dimension apart from those objects (Grant 551). Aristotle considered this theory silly, believing that if place were itself three-dimensional, it would be a thing. He argued that “place cannot be body; for if it were there would be two bodies in the same place”—that is, any given object would interpenetrate the body of the place it occupied (Phys. IV.1, 208b 5). Instead, Aristotle defined “place” as the “innermost motionless boundary” of whatever body contained it (Phys. IV.4, 212a 20).
On the other hand, Duns Scotus argued that void space itself moved aside for bodies when the bodies moved into it—and that the place of something could be determined by either a motionless or moving container (Grant 564, citing Duns Scotus 300). So, for example, a boat anchored in mid-stream would be changing place constantly as the water flowed around it (Ariew 437).
The Proem to Book V of The Faerie Queene looks for something like Aristotle’s “innermost motionless boundary,” a set of coordinates by which it would be possible to measure the place of the entire world. The speaker is disturbed at the changes the world has undergone, yet the first metaphor he gives for this disastrous change represents an attempt to contain the disaster by plotting it geometrically: “Me seemes the world is runne quite out of square / From the first point of his appointed sourse” (V.Pr.1.7-8). The figurative meaning of “square,” which had become popular in the sixteenth century, is uppermost here: the world has fallen into disorder. But figurative uses of the phrase “out of square” strongly evoke the phrase’s literal origins: the carpenter who finds an old joint out of square knows that the larger structure has been warped and weakened. The carpenter must then figure out how to alter the warped geometry to bring it back into right-angle planes. When, three stanzas after the lines quoted above, the speaker of the proem observes anxiously that “the heauens reuolution / Is wandred farre from, where it first was pight,” he refers to a problem with the celestial coordinate system, which was apparently going out of square (V.Pr.4.6-7). In order to calculate the position of a planetary body, astronomers started with the intersections of certain right angles and circles, as projected from a point on earth’s equator during the vernal equinox. What disturbs Spenser is a phenomenon observed but imperfectly understood by Ptolemy and other ancients: that over thousands of years, the equinoxes had slowly precessed, moving backwards across the sky, with the result that signs of the zodiac did not plot correctly. The celestial grid of latitudinal and longitudinal lines seemed to distort.
We now know that this slow alteration of coordinates is caused by a wobble of the earth’s axis over a 26,000-year cycle, but to earlier minds it seemed like a disturbing deformation of God’s original design. Topologically speaking, it was as if the planets and stars were points on constellation-shapes that had warped over time, the way a rectangle drawn on paper can be twisted and folded. “And so,” mourns the speaker, the out-of-kilter heavens “doe make contrarie constitution / Of all this lower world …” (V.Pr.4.8-9). What we used to call virtue is now called vice, and everything on earth is changed for the worse. The celestial distortions cause a warping of human history, of individual character, of social institutions, and of civil structures. Words themselves—“truth,” “justice,” “virtue”—lose their reference points. At the same time, the poet’s conception of the world’s moral degeneration as the result of an astral crookedness that could theoretically be visualized and measured from a particular vantage point on earth makes the human disaster at least potentially less incomprehensible.
In the proem’s second stanza, the speaker conceives moral change in terms of the transformation of matter: the golden age on earth has become a stony one, and the substance of human bodies has changed from flesh into stone. Spenser’s language is consciously scientific: we read that men “at first were framed / Of earthly mould, and form’d of flesh and bone” (V.Pr.2.3-4). In Aristotle’s Physics, bronze is the matter that can be made into a statue, and wood is the matter that can be made into a bed, so flesh and bone are the matter that, when given form, make a human being (Phys. I.7, 191a 10; II.2, 193a 30-193b 1; II.2, 194b 9). Spenser’s language hearkens back to the philosophical world of the Garden of Adonis, but here in Book V the formation has become creepy. It is like one of those films in which people realize that a friend has gradually, almost imperceptibly, been replaced at the cellular level by alien matter. (The less anachronistic analogue is, of course, Ovid’s Metamorphoses, though it is difficult to know whether its depictions of humans changing into other forms of matter were as disturbing to Spenser’s contemporaries as film depictions of alien invasion are to us.)
According to Aristotle, the transformation of one substance into another—as when an animal eats food—is not evidence of interpenetration, because one body is disappearing as fast as the other is appearing. When an animal destroys plant matter to generate new flesh, the total amount of matter remains the same: “It is not a growth of either” (Aristotle, Gen Corr. 321a 1-10). Despite Spenser’s scientific language, however, the fact that we know, looking around us, that human beings are still formed of flesh rather than of stone means that the analogy asks us to imagine two bodies in the same place at the same time—to imagine that the bodies of human flesh that we see around us are simultaneously completely stone. And although this is physically impossible, it is also fitting, because as the poem has demonstrated over and over, admirable selves and regrettable selves frequently interpenetrate.
The proem’s anxieties about physics gone awry reach maximum intensity in the lines in which we are told that the constellations have inexplicably occupied each other’s places, each one shoving another aside:
So now all range, and doe at randon roue
Out of their proper places farre away,
And all this world with them amisse doe moue,
And all his creatures from their course astray,
Till they arrive at their last ruinous decay.
When we and the other creatures arrive at our destination, it will be no place at all. This is a journey from form to formlessness, with no prospect of rejuvenation or reformation.
In canto 2, after Artegall and Talus have “refourmed” Munera’s realm (28.8) by doing their best to remove it physically from the face of the earth, they meet the giant with the scales—who is notorious among Spenserian scholars for being punished when he expresses fears that seem quite similar to those expressed in the proem (V.ii.28). Every element, the Giant says, has encroached upon every other element. The sea has taken from the land, “And so were realmes and nations run awry. / All which he vndertooke for to repaire, / In sort as they were formed aunciently” (V.ii.32.6-8). Artegall responds by giving the Giant a physics lecture that registers a touch of hysteria. If the speaker of the proem longed for innermost motionless boundaries, Artegall claims that he has found them. Everything in the world was put in place by the Creator:
The earth was in the middle centre pight,
In which it doth immoueable abide,
Hemd in with waters like a wall in sight;
And they with aire, that not a drop can slide:
Al which the heauens containe, and in their courses guide.
He declares that “euery one doe know their certaine bound” and that “mongst them al no change hath yet beene found” (V.ii.36.2, 4). The latter statement does not necessarily contradict the proem, given that the Giant is concerned about the ratios among quantities of matter. Like the philosophers, Artegall avers that the sum total of matter does not change in quantity: when the waves eat at the land, they do not actually reduce the earth’s total amount of soil, because the tides deposit that soil elsewhere.
But it is cold comfort to tell a nation that the earth’s sum total of land remains the same when some of the nation’s land has been carried by the tides to another nation’s shore—or when a foreign army has claimed some of that nation’s land and transformed it into another political substance. What does “place” mean then?
In Book V, Spenser asks whether it is possible for one nation to penetrate another without first obliterating it—that is, whether nations can ever interpenetrate while conserving cultural matter. He examines this question at least metaphorically with the apparatus of science, which suggests that he wants precise answers. And yet while both the proem and Artegall’s lecture to the Giant demonstrate the author’s investment in believing the philosophers’ conclusions that matter is conserved and bodies are impenetrable, Spenser also seems heavily invested in disbelieving physics when it is more convenient.
On the one hand, Talus is said to be “Immoueable, resistlesse, without end” (V.i.12.7). Like the truth that he is said to “vnfould” with his flail, Talus gradually spreads out to occupy all foreign space. Sometimes he kills off his foes and their followers, but at other times, the followers simply disappear, and we hear no more about them. Do Munera’s loyal subjects perish with her castle? Do the Giant’s groupies ever emerge from their hiding places in the bushes? It does not matter to the narrator; their bodies cannot occupy the same place that Talus occupies.
On the other hand, the instant disappearance of a conquered people and the instant conveyance of their substance to their conquerors is wish-fulfillment of the least likely kind, so Book V—like A View of the State of Ireland—poses alternative scenarios. When a part of the English body politic locates itself abroad, it may transform its substance into foreign substance, as Irenius accuses the Old English of doing in A View of the State of Ireland (a transformation that is not wish-fulfillment for Spenser’s English readers), or it may change foreign substance into its own substance (better, from the English point of view), or it may displace foreign bodies (not bad, though a good deal of energy must be expended in keeping them displaced), or it may occupy any of a number of places whose only analogies in the language of physics are impossible ones.
It is especially fitting that the character Belge has a problematic body, given that she represents a country without a head—a land reclaimed uneasily from the sea and occupied by foreign powers. Below are the lines in which the narrator describes the Spanish Duke of Alva’s occupation of Antwerp and his building of a citadel on the River Scheldt in 1567 to keep the Dutch under Spain’s thumb. The first pronoun, “They,” refers to Arthur, Belge, and two of her sons (Holland and Zealand). Pay attention to the remaining pronouns:
They came vnto a Citie farre vp land,
The which whylome that Ladies owne had bene;
But now by force extort out of her hand,
By her strong foe…
So far, “her” refers to the Lady, Belge.
… who had defaced cleene
Her stately towres, and buildings sunny sheene;
Shut vp her hauen, mard her marchants trade,
Robbed her people, that full rich had beene,
And in her necke a Castle huge had made,
The which did her command, without needing perswade.
Reading that the Duke of Alva erected a castle in the neck of the River Scheldt is quite different from reading that he erected it in Belge’s neck. One could conceivably argue that the pronoun “her” now refers to the City of Antwerp, but there has been no marker of such a transition. Only retroactively might we decide that perhaps “Her stately towres” meant those belonging to the City rather than those belonging to Belge, thus allowing all of the pronouns from then on to refer to Antwerp—but even retroactively, there is little to support that interpretation until we get to the castle in “her” neck. The image of a castle sticking out of a woman’s neck conveys the grotesquery of the Spanish occupation of Antwerp, but along with the earlier stanzas about her banishment to the marshes, it also gives Belge a topologically impossible body.
When Arthur first meets Belge, the Tyrant Geryoneo has taken over her land and has banished her to the marshes: “Ay me (sayd she) and whether shall I goe? / Are not all places full of forraine powres? / My pallaces possessed of my foe, / My cities sackt …” (V.x.23.1-4). If the narrative clearly made Belge a queen, it would make complete sense that an invader could dispossess her of her land. But the Low Countries had neither queen nor king, the Prince of Orange being the nearest thing to a monarch, and Spenser refers to Belge only as a lady. She would thus seem to represent the Low Countries’ body politic without a monarch’s body natural. That would make sense, given that as many as half the people of Antwerp did, in fact, flee the city after Philip II recaptured the city in 1586. And yet because the narrative mentions Belge’s people and her state—and indeed, her cities—as entities that are separate from her, Belge cannot quite be the body politic, either. What is left for Belge to represent but the land itself, the geopolitical concept called the Low Countries?
Insofar as she is either the land or the body politic, Belge has been driven out of herself. She is solitary, without her people, living apart from the place that erstwhile defined her. What is she, without a place? Although the concept of the Klein bottle was not developed until 1882, one cannot help thinking of its three-dimensional topological surface, which cannot be oriented and whose inside is indistinguishable from its outside. Belge, like a Klein bottle, is a shape without a boundary—not because she is limitless but because she cannot be oriented.
The Faerie Queene ultimately depicts a somewhat rosier picture of Belge’s rescue than what actually transpired in Spenser’s lifetime. Spenser wants to believe that national boundaries can be defined and, indeed, that they have already been divinely defined, yet the impossible topology of Belge’s body suggests otherwise.
By the end of Book V, when we get to Envie gnawing out her own bowels—another Klein bottle image—and mentally join that image with those of earlier books—Alma besieging herself, Scudamour trying to gain entrance into the house of his own fears, Ate’s castle being both a place and (disastrously) all places, and so on—we can see that for Spenser, place is almost inevitably bound up with movement and with a lack of orientation. A nation is like Duns Scotus’s boat, its place defined by the water that moves with it downstream.
Considering Spenser’s interest in the relationship between national placement and toplogical deformation, it is significant that all six occurrences of the lemma “reform” and its variations in The Faerie Queene take place toward the end of Book IV or somewhere in Book V. In the ninth canto of Book IV, we read that Poeana has “reformd her waies” (IV.ix.16.8). In the first canto of Book V, Sanglier threatens to reform Artegall in a fight (V.i.21). In canto two, Artegall and Talus reform Munera’s customs by cutting off her hands (V.ii.28.8). In canto ten, the narrator tells us that it is “better to reforme then to cut off the ill,” thus calling into question the efficacy of Artegall’s and Talus’s supposed reformation of Munera’s body politic (V.x.2.9). In canto twelve, variations of the word “reform” are used twice in reference to Artegall and Talus’s desire to heal Ireland by killing all disobedient Irish (V.xii.26.4, xii.27.1). These last two references to reformation hint at an impossibility that, in hindsight, has gripped the whole of Book V: Irena can be Eirena only insofar as she has no body politic. Like Irenæus in A View of the State of Ireland, she exists only by being absent from herself.
In Book V, then, the word “reform” is always a pun on giving a politicized body new physical form by violence—and none of these reformations turns out satisfactorily. Insofar as Book V considers the interpenetration of English culture and that of Ireland or the Netherlands in terms of spreading and strengthening the Protestant form of Reformation, prospects are not good.
Marvell, too, uses the physics of bodies to conceptualize the place of nations—though Marvell rather enjoys disorientation in his spare time, if “Upon Appleton House” is any indication. In that poem, his narrative eye takes delight in swooping abruptly from the perspective of grasshoppers below the grass to that of someone looking through a telescope at the wheeling stars, and so on. Even in a poem as politically vituperative as “Fleckno, an English Priest at Rome,” Marvell revels in imagining political foes as impossible bodies. I have mentioned the moment on Flecknoe’s staircase when the narrator makes a metaphysical joke about three bodies interpenetrating. There is another lovely moment—or actually, quite an unlovely one—in which the narrator describes the ascetic Flecknoe as being so thin that he must put on his clothing to give himself a body:
Nothing now Dinner stay’d
But till he had himself a Body made.
I mean till he were drest: for else so thin
He stands, as if he only fed had been
With consecrated Wafers: and the Host
Hath sure more flesh and blood then he can boast.
But clothing is not enough; Flecknoe stuffs his coat with the paper on which he has written his bad poetry:
Lest his too suttle Body, growing rare
Should leave his Soul to wander in the Air,
He therefore circumscribes himself in rimes;
And swaddled in’s own papers seaven times,
Wears a close Jacket of poetick Buff,
With which he doth his third Dimension Stuff.
A few lines later, the narrator describes Flecknoe as a “half-transparent Man” (80). Flecknoe almost does not occupy space. To the narrator’s great amusement, the Catholic priest comes perilously close to being the void space the existence of which was denied by Aristotle.
In “An Horatian Ode Upon Cromwel’s Return from Ireland,” the speaker grudgingly and ambiguously praises Cromwell for having taken the place of King Charles:
And, if we would speak true,
Much to the Man is due.
Who, from his private Gardens, where
He liv’d reserved and austere,
As if his highest plot
To plant the Bergamot,
Could by industrious Valour climbe
To ruine the great Work of Time,
And cast the Kingdome old
Into another Mold.
And a few lines later:
Nature that hateth emptiness,
Allows of penetration less:
And therefore must make room
Where greater Spirits come.
Cromwell is not moving into an empty space; instead, he is taking over the matter of Charles’ body politic and re-forming it—and not necessarily for the better, given that the previous form was “the great Work of Time” (34). If Artegall were here, he might give Cromwell the lecture he gave to the Giant: it is perilous to change the great work of time by arbitrarily reassigning matter and nations.
The speaker compliments Oliver Cromwell by saying that Cromwell is too ambitious on behalf of his own country to want to stay in one place long and that he is too courageous to tolerate anyone’s attempt to enclose him. He compliments Cromwell by saying that the hero has left the comfort of his country property out of a sense of duty, and by saying that like the falcon who does not rest until it has brought the kill back to its master, Cromwell does not rest until he brings Ireland as a prize back to the English Commons and presents it to them “for his first years rents” (86). Instead of seeking the post of Lord Protector with the hope of personal gain, he merely rents that place, on behalf of the people. Yet these compliments also hint that Cromwell occupies the place of Lord Protector uneasily and even that the temporary nature of his tenure is fortuitous:
So restless Cromwel could not cease
In the inglorious Arts of Peace,
But through adventrous War
Urged his active Star,
And, like the three-fork’d Lightning, first
Breaking the Clouds where it was nurst,
Did thorough his own Side
His fiery way divide.
There is grandeur in the image of Cromwell as lightning breaking through the clouds of his own political party, his fellow Parliamentarians who at first opposed him, yet the phrase about his dividing his fiery way “thorough his own Side” inevitably calls up a less grand, and in fact grotesque, mental image: that of Cromwell bursting out of his own body.
He does not even properly inhabit his own body. It is true that his breaking out is a personal sacrifice, since he would rather stay home and garden, and Marvell certainly does not seem averse to Cromwell’s having “tam’d” the Irish (74) or to the prospect of his putting the Scots into their proper place. Yet the poem’s celebration of Cromwell’s constant movement from place to place also makes it clear that this is not the sort of imaginative movement celebrated in “Upon Appleton House.” Instead, it is an austere duty. And something curious happens when the speaker finishes talking about Ireland and turns to Cromwell’s next project, that of annexing Scotland. This bit comes right after his praise of Cromwell for having dutifully turned over Ireland to the Commons:
So when the Falcon high
Falls heavy from the Sky,
She, having kill’d, no more does search,
But on the next green Bow to pearch;
Where, when he first does lure,
The Falckner has her sure.
What may not then our Isle presume
While Victory his Crest does plume!
What may not others fear
If thus he crowns each Year!
Next the speaker compares Cromwell to great generals of the ancient world. Like the Gallic tribes when Julius Caesar invaded their territory and like the Romans when Hannibal invaded theirs, the Picts (the poem’s derogatory term for contemporary Scots) will be unable to escape Cromwell’s advance: “The Pict no shelter now shall find / Within his party-colour’d Mind” (105-106).
Cromwell will displace the “Pict” from himself—as Geryoneo did to Belge in The Faerie Queene—and though the image of the Pict’s being prevented from taking shelter in his parti-colored mind indicates that by nature the Pict normally does take refuge in a treacherous changeability, these lines also oddly align the Pict with the speaker of Marvell’s poem “The Garden,” which was probably written in the same year as the Ode. The speaker of “The Garden” withdraws first into the solace of a lovely garden and then into the solace of his own colorful mind, which is at once narrower and more spacious:
Mean while the Mind, from pleasure less,
Withdraws into its happiness:
The Mind, that Ocean where each kind
Does streight its own resemblance find;
Yet it creates, transcending these,
Far other Worlds, and other Seas;
Annihilating all that’s made
To a green Thought in a green Shade.
Here at the Fountains sliding foot,
Or at some Fruit-trees mossy root,
Casting the Bodies Vest aside,
My Soul into the boughs does glide:
There like a Bird it sits, and sings,
Then whets, and combs its silver Wings;
And, till prepar’d for longer flight,
Waves in its Plumes the various Light.
The “Ode Upon Cromwel’s Return from Ireland” may not register full-blown regret at the impending colonization of the Pict’s colorful mind, but certainly Marvell’s portrait of Cromwell gives no evidence that Cromwell has the sort of colorful imagination that Marvell admires. The poem calls Cromwell austerely brave, strategic, heavily effective, and industrious (30, 48, 92, 33). Commending someone for being “industrious” is almost as bad as saying that he does well under supervision. Oh—and then the speaker tells Cromwell that the Scots are waiting to be conquered by him: “[T]hou, the Wars and Fortunes Son / March indefatigably on” (113-114). That is just what a war hero wants to hear: “Sir, you are excellent at marching.” However commendable indefatigability may be, it is hardly inspiring. The territory inside Cromwell’s mind is not a place to which Marvell would want to retreat.
Like Spenser, Marvell has a poem about the Low Countries, which by now were free from Spain. England was now livid at them for encroaching on English shipping, resulting in naval battles. In Marvell’s nastily witty poem “The Character of Holland,” the Dutch body politic is all matter—a lump—and therefore occupies space, but no space in particular. It has no set form, nor even a set amount of matter. Like the animals whom Aristotle mentions as converting food into their own flesh and therefore conserving the total amount of matter, Holland takes matter from another when it borrows mud from England to pile up for its own land—but unlike Aristotle’s animals, Holland is incapable of successfully converting that matter completely into itself. It “scarce deserves the name of Land,” because it is made of the “indigested vomit of the Sea” (1, 7). The sea regularly displaces Holland, at which times fish take the place of businessmen. The Dutch take a dried salt-fish as their Protestant savior.
Furthermore, instead of confidently occupying their place as Protestants, the Dutch have allowed their religious culture to interpenetrate with all others: “Turk-Christian-Pagan-Jew / … where not one so strange / Opinion but finds Credit, and Exchange” (71-74). Yet the poem hints that the Dutch citizens’ status as matter without form will make them more difficult to conquer:
See but their Mairmaids with their Tails of Fish,
Reeking at Church over the Chafing-Dish.
A vestal Turf enshrin’d in Earthen Ware
Fumes through the loop-holes of a wooden Square.
Each to the Temple with these Altars tend,
But still does place it at her Western End:
While the fat steam of Female Sacrifice
Fills the Priests Nostrils and puts out his Eyes.
The Dutch women in church place their bums on stoves full of smoldering Dutch turf which is simultaneously English turf, the heat from which disperses the women’s bodies throughout the room in the form of a choking vapor that nonetheless resists political sublimation. If Holland’s land is difficult for the Dutch to fix in place, Holland’s body politic will also be difficult for the English to grasp. It travels, it resists form, it is no place. For Marvell, as well as for Spenser, “place” is inseparable from motion and is inevitably bound up with impossibility—which raises problems for their dreams of political occupation.
University of Arkansas
 All citations of Marvell’s poems refer to The Poems and Letters of Andrew Marvell, ed. H. M. Margoliouth, rev. Pierre Legouis and E. E. Duncan-Jones, vol. 1 (Oxford: Clarendon, 1971).
 All citations of Aristotle’s works refer to Complete Works of Aristotle: The Revised Oxford Translation, ed. Jonathan Barnes, vol. 1 (Princeton: Princeton UP, 1984), Bollingen Series 71.2.
 Edward Grant, “The Principle of the Impenetrability of Bodies in the History of Concepts of Separate Space from the Middle Ages to the Seventeenth Century,” Isis 69.4 (1978): 551-571, and John Duns Scotus, Quaestiones in Lib. II Sententiarum, d. 2, q. 9., Opera omnia VI.1. (Hildesheim: Georg Olms, 1968) 300.
 Roger Ariew and Alan Gabbey, “The Scholastic Background,” The Cambridge History of Seventeenth-Century Philosophy, ed. Daniel Garber and Michael Ayers, vol. 1 (Cambridge: Cambridge UP, 2012) 423-453.
 All citations of Spenser’s Faerie Queene refer to the Suzuki edition, The Faerie Queene, ed. Toshiyuki Suzuki, Shohachi Fukuda, A. C. Hamilton, and Hiroshi Yamashita (Toronto: Pearson Canada, 2006).
 In mathematics, topology is the study of the “properties of figures and surfaces which are independent of size and shape and are unchanged by any deformation that is continuous, neither creating new points nor fusing existing ones; hence, with those of abstract spaces that are invariant under homoeomorphic transformations” (“topology, n.,” OED n.3.a., OED Online, Oxford UP [June 2015]).
 The best visualizations of a Klein bottle I have found are on the Plus Magazine website, part of the Millenium Mathematics Project of Cambridge University. See the first two animations at “Introducing the Klein Bottle” and the first animation at “Imagining Maths—Inside the Klein Bottle” (accessed August 30, 2015). Although it is possible to construct a physical approximation of a Klein bottle—say, out of glass—the mathematically-conceived Klein bottle cannot take physical form, because it does not intersect itself at the point where its narrow tube enters the wider cone. If you look closely at the animation posted on the second link above, you will see that at one point the arrows travel along what intuitively seems the “outside” of the tube, right through the side of the larger cone, with no barrier. (Online viewers who insist that a Klein bottle has internal volume are overlooking the difference between physical representations and the Klein bottle as mathematically conceived.) For an illustration of why some shapes are defined as non-orientable, see the animation at the end of the first article above, depicting a face traveling around a Möbius strip.